KIVA-II: a Computer Program for Chemically Reactive Flows with Sprays

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LA-11560-MS

KIVA-II: A Computer Program for Chemically Reactive Flows with Sprays

Los Alamos N A T I O N A L L A B O R A T O R Y

Los Alamos National Laboratory is operated by the University of California for the United States Department of Energy under contract W-7405-ENG-36. Prepared by Adrienne Rosen, Group T-3

This work was supported by the US Department of Energy, Division of Energy Conservation and Utilization Technologies.

An Affirmative Action/Equal Opportunity Employer

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither The Regents of the University of California, the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by The Regents of the University of California, the United States Government, or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of The Regents of the University of California, the United States Government, or any agency thereof. Los Alamos National Laboratory strongly supports academic freedom and a researcherÕs right to publish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness. LA-11560-MS

UC-96 Issued: May 1989

KIVA-II: A Computer Program for Chemically Reactive Flows with Sprays A. A. Amsden P. J. OÕRourke T. D. Butler

Los Alamos N A T I O N A L L A B O R A T O R Y Los Alamos, New Mexico 87545 TABLE OF CONTENTS

ABSTRACT ...... 1

I. INTRODUCTION AND BACKGROUND ...... 1 n. THEGOWRNINGE QUATIONS ...... 7 ~, Tk-e-~5.kic2Phase==..=, =. =. ==. . ...=...... T IL. ‘[email protected]@ets-:..=..=...... 12- C.. Boundary-Conditions ...... ,.,.:.=.=.=...... =.. 20. m. THEN~QT3mRIcALs+IwFim= ...... ~q= A. Telmporal”D”iffweneing ...... 24- B. Sp_atial Differencing ...... ~: .-.. -.. =–.,–..–. ...=.. ...’...... 24 StOc-ha-sti?Parti5ie’~e&nique ...... 32. D; State IReiations ...... 34 E. ~~~dII~fid~-~h”aS~ ~iI~erenCe ~qUatiOns ... ,...... 34. Mass Density Equations ...... 34 ;: MomentumEquation ...... 36 3. C41Fa~h T-ormall#eloeities- ...... —...—,—,—,—,—=—=—3-7 4.. bterndfier~.Eq.ution .,...... 39- 5; T--ur*ulen~e-Equa~i~n~- ...... —.—,—,—,—,—,—:—41 6. VolumeChang&Equation and Equations ufState ...... 42. P ~tiopkt-~quatiims...... 43. ~. $olutionP>oeAur~forbplieitPhase B-~@uatiom...... 43- G. Phase C ...... 47 H. Accuracy Conditions and Automatic Timestep C&nt;~i ...... 52 12v-.. TEECICIWH!l/%31L~OGRAM ...... W- _A”.- (@nera~ S&ru@~re ...... —,—,.,.,.,.=.:...... ~&. B-. The Computing Mesh ...... 60 L T-heFiv@&sh .T=y~*-*ailab}e. . . ~: ...... –:–,—,—=.=.=...... ~& 2-D “tO3-D(jjnv~r~ion .-.-..–-–-–=–~–=–=–.–=–.–.–.–.–.—.–.=.–==.=..=..=.. 53 c. The Indexing Notation ...... ~-.l-..~”..~:.lL1 63 D. &orageolZel]”13ata ...... 64 ~. Me~hGene~+,ion ...... ~q= 1. T~e15stm-.Face ...... 08P. 2. The Cylinder Head ...... ::::::::: 74 -- J!. Celland Vertex Flags ...... 76 G. F-uelSprays ...... 77 Spray. Origjn, Profile, and Orientation .,, . . . .. s..=..,.. =..=....’ ....’. ~~ ;: Spray Flow Definition ...... =. . =...... ‘?9 ~:3. Particle Radius ...... 81 stockiastiikjdjo~...... S1 H; Spar!! Ignition=...... 81 L. kitial.Be~elElmcti~ nSWirlProfiie ...... Om- J-. Fuel Library ...... --.-.-.-.,.,.=.=.=.=.=.=.-.---.-....=.,z= K: LaP.-ew.andGutfiow.Boundaities- ...... 85”- L. output ...... 87 M. ~WS~@tiV~-~~OtS...... 88 N. Chopper ...... –.–.—=—=—=—=—=–=–=—=—=—,...... 94. 0. Ikq22m~Rf#x3r~...... 94 ACKNOW-LEDG~NTS ...... 95

v A.PPENDIX A. DETERMJ.NATION OF THE PGS PARAMETER ...... 96

A.PPENDIX B. TURBULENT BOUNDARY LAYER TREATMENT ...... 98

APPENDIX C. NUMERICAL SOLUTION OF THE EQUATIONS GOVERNING SPRAY DYNAMICS ...... 107

APPENDIX D. PARTICLE RADIUS SELECTION AT INJECTION ...... 113

APPENDIX E. THE DROPLET COLLISION CALCULATION ...... 115

A.PPENDIX F. THE DROPLET OSCILLATION AND BREAKUP CALCULATION ...... 117 APPENDIX G. CALCULATION OF DROPLET TURBULENT DISPERSION 120

APPENDIX H. THE VARIABLE IMPLICITNESS PARAMETERS ...... 122

APPENDIX I. KINETIC CHEMICAL REACTIONS ...... 128

AJ?PENDIX J. EQUILIBRIUM CHEMICAL REACTIONS ...... 129

APPENDIX K. CALCULATION OF VELOCITY GRADIENTS AND VISCOUS STRESSES ...... 136 APPENDIX L. THE ALTERNATE NODE COUPLER ...... 137

APPENDIX M. QUASI-SECOND-ORDER UPWIND DIFFERENCING . . . . . 142

APPENDIX N. PARTIAL DONOR CELL DIFFERENCING ...... 149

APPENDIX O. ROBIN HOOD ALGORITHM ...... 151

APPENDIX P. ANGULAR MOMENTUM CONSERVATION LOGIC ...... 152

REFERENCES ...... 155

vi KIVA-11: A COMPUTER PROGRAM FOR CHEMICALLY REACTIVE FLOWS WITH SPRAYS

A. A. A-msden; P; J. (3’Ruurke; and--T. II: i3utier

ABSTRACT

T.hi~rePor~- documents the-K! VA--11-computer-pro-gram for the numerical calculation of transient, two- and three-dimensional? chemically- reactive- tl-uid- fl-ows with sprays. K-IVA-H extendsand: enii~ the-earlier. KI..V..cde,e, impro%’-ing-~t%eomputatio nal -aam- racy and efficiency and its ease-of-use. The KIVA-11 .equatiuns.andnumerical solution procedure are very general and can be applied to. ia~ina~ oc tur-bulent- flow% subsonic-or supersonic- flows; anti- skgk-pka-o-r dkqxrrsed-- two-phase fio ws. A-rbitrary numbers of- speck and chemica.r.etikms .ars -ak-wda.. .A4KM4R3*% gm-%ie+e method is used to calculate evaporating.liquid sprays, includingthe- effects of-”di-oplet-collisions and aerodynamic breakups. Although the initial and boundary conditions and mesh generationhamxbeen- written for internal combustion engine calculations, the logic for tlmse- specificatio-ns- c-an- be easily modified: for a variety of- oth-er applications= Eolluwlng--an=o\’ecview of-ihe=pl+uci~l=tiiul-es-of=the KIVA-11 program, we describe. in-detail. the equations- solved$ t,h~ numer-ical solution- procedure, and- the- structure of the computer pr~gr-am, Six%een -appendices provide additional- cietails co.nccnm- ;ngthe -numetical :s0Iuti unqxa-ce-d ure.

1. INTRO DUC-1’IW.AND.BACK.GROUND-

The in-cylinder dynmicsoftianc~d ~nternal.mbustion en@ne~j saeh as--the direct-injection stratified-ehargdDISC) engine, involve anumber&compkx, closely ~mupledphysic%l and-chemical processes These include &lietransiimt three-dimensional t @n=~m .u~-evapra%ifigf. e tiei’sprays i“nteractfng witii fIowing multicomponent gases undergoing inixing; ignition; chemical reactions, and “heat transfer.. ‘The KIYF.-codei ‘J“has k the ability to calculate such flows in engine cylinders with arbitrarily shaped piston L geometries, including.the.effeckcdkmhulence and wall heat transfer. In response to the L needs of a kirg~- user community and to reeentdeveloprnents imthefieids-of -numerical fluid dynamics -mdinternal combustion engjne mocieiing> we ‘kiave-irnpkmemted .many. E improvements to ‘KIVA- .since.ikpuhlk release in 1985. T-hethan ggs~e&cor~ratedin a ~ 1 new version of the code, called KIVA-11, that is documented in this report. KIVA-11 builds on the capabilities of I-UVA and is quite similar in structure. Current users of KIIVA will find the transition to KIVA-11 to be straightforward. An excerpt from Ref. 1 explains the basis under which IS(VA was written: ‘engines in mind, it contains several features designed to facilitate such applications. However, the basic code structure is modular and quite general, and most of the major options (chemical reactions, sprays, etc.) can be individually activated or deactivated by setting appropriate values for the associated input switches. The code is therefore applicable to a wide variety of multidimensional problems in fluid dynamics, with or without chemical reactions or sprays.” Indeed, KIVA has been used for numerous studies besides internal combustion engines, including cold flow analyses in complicated geometries, continuous spray combustors, Bunsen burner flames, nonreacting sprays, and hydrogen-oxygen flames propagating in long tubes, to name just a few. It is impractical to cite all such studies here because of the widespread distribution and use of the code in industry and universities. For internal combustion engines, besides the studies of the DISC engine that have been carried on at General Motors Research Laboratories, Princeton University, and Los Alamos National Laboratory,4-7 it has been used as the basis for numerical investigations of diesel engines8-10 and of coal-fired dieselsl 1as well. From a historical perspective, KIVA-11 is the latest in a series of multidimensional codes that we have produced since we began work on numerical simulations of internal combustion engines 12 years ago, under the sponsorship of what has become the Department of Energy’s Energy Conversion and Utilization Technologies (ECUT) program. All of them are multidimensional finite-difference codes that solve the transient equations of motion. The first of these was the RICE code. 12 RICE was a two-dimensional Eulerian code that utilized rectangular computing zones for its mesh, eddy diffusivity to model the turbulence, Arrhenius kinetics with an arbitrary number of reactions and species to represent the chemical kinetics, and a partially implicit finite difference formulation to efficiently treat the acoustic terms for low Mach number flows. Bracco et al. at Princeton modified RICE and produced the REC code,*3 which included the effect of piston motion in the unresolved third dimension of the calculations. Another two-dimensions Eulerian code, APACHE,14 followed RICE. This had the capabilities of RICE and the generality of arbitrarily shaped cells. CONCHAS15 followed APACHE, and it likewise utilized arbitrarily shaped cells but offered the feature of an arbitrary Lagrangian- Eulerian formulation that allowed the computing zones to follow the piston motion. In addition the turbulence effects were included in the calculations by use of a subgrid scale model. CONCHAS-SPRAY*G replaced CONCHAS. As its name implies, it included a model for the spray dynamics, a statistical representation that accounted for a spectrum 2 of droplet sizes and the effects of evaporation. The turbulence was calculated by means of u-ubgrid-s~de -model that used a transportequa-tion for turbulent kinetic energy anda kwv-ef-the-wall .treatientfo~turbulent Emm&dry ‘layers; ‘Me-chemistry was generalized” to irdudeboth kinetic and equilibrimrewtim~-~-A1’2 the.n&dliMvecl.In addi tion=to= retaining the capabilities oFC”O-N--C-H-AS-b~AY,it featured the ability to do either two- or three-dimensimmdl problems-with the same code. Furthermore, it had “anexpanded- spray. nmdel tbat:&~*+s=;lli&c=s-+mdatitil differences-a-r~fomed ona finite-difference mesh that subdivides the computational region into a number of small cell~thtare hexahedrons. T-he corners of the cells-arecalled vert,iee~ -~ti.bepositia~~s-of= the vertices may. be.mbitr=ily.spetified.functi~ns of=time, thereby, allo}ving=a-La~an@ =;. Euierian j or mixed”d~seri”ptiafi.-— ‘Tlie arbitrary mesfictinco~titimzh:cu~ed boundaries andcan-~movetofullowcham-ges in combustion chamber geometry. A strength oftlie method”is that the mesh need-no$ be orthogonal. The spatial differeneingis- made conservative wherever possible. The procedure used is to difference the basic equations in inte, gal form, w-ith the volume of a typical ceil used-as-the control volume, and with divergence -terw.atraraf6rme&to&f%c&i+ite~als utingthe divergence theorem.;$ The Cartesian components of the velocity vector are stored at cell vertices, and the momentu~n-e~ations-a~e-d ifierenced” in a strict]~ conservative tashlon. fi c.ontr.asLti tthe original ALE method, 17’18however, cell. faced velocities are used during a portion of the mrnputa-tion al -eye}e.2 Their use -greatly mx-lucesthe-temiency of the A-LE method “to p.ara- sitic velocity modes, .tl-mrebylarge.iy.elimindhg th&nfHLffQr.rlQde CcMq3k. Tiie transient soitition is marched-out in a sequence of finite time increments called cycles or timesteps. On each cycle the values.of.the dependent variables are cakmlatei

3 TABLE I

KIVA-11 FEATURES

1. Computational Efficiency Improvements . Coupled, implicit differencing of diffusion terms and terms associated with pressure wave propagation . Subcycled calculation of convection ● Stochastic spray particle injector . 2-D to 3-D converter

2. Numerical Accuracy Improvements ● Optional quasi-second-order upwind convection scheme . Generalized mesh diffusion algorithm . Method for computing turbulent droplet dispersion when At exceeds turbulent correlation time . Convection of length scale in place of the turbulence dissipation rate e

3. New or Improved Physical Submodels . k-g turbulence model . Model for droplet aerodynamic breakup

4. Improvements in Ease-of-Use and Versatility Nonflat cylinder head option Inflow/outflow boundaries Simplified velocity boundary conditions Alphabetized epilogue listing FORTRAN variables and their definitions Gravitational terms Eulerian and Lagrangian options Library of thermophysical properties of common hydrocarbons Initial Bessel function swirl profile Optional tabular input of spray injection velocity from those on the previous cycle. As in the original ALE method,17*18each cycle is divided into two phases - a Lagrangian phase and a rezone phase. In the Lagrangian phase the vertices move with the fluid velocity, and there is no convection across cell boundaries. In the rezone phase, the flow field is frozen, the vertices are moved to new user-specified positions, and the flow field is remapped or rezoned onto the new computational mesh. This remapping is accomplished by convecting material across the boundaries of the computational cells, which are regarded as moving relative to the flow field.

4 Immntrast.to.KIVA, the temporal difference scheme in mA-~ is largely implicit. Because of this, the timesteps used by KIVA-11 are calculated based on accuracy, not stability, W3‘ixwii+andean-be considerab]~’-~arger.than.the t.iinesteps Jdsed.by.~-A-.. T!is has resulted in considerable savings of computational time in many problems. In the Lagrangian phase, implicit differencing is used for all the diffusion terms and the terms associateibvith pressure wave propagation. me coup~~d-i-rnpi~~ltequatl~ns are so~ved”b~ a method’sirniia~ ti-the-SI?@LE-l Q’algotithm; w-itb-individual:equations being solved-by the conjugate residual method.20 Explicit methods are used to-calculate convection in the rezone phase; but the convection calculation can be subcycled an arbitrary number of times, .and.thus the.main. cmnputationa-1 timeskep iS not restricted by the Courant stability ~mndition of explicit methods.21 The convection timestep is a submultiple of the main computational time-step and does satisfy the Courant condition. In addition to the partial donor cell-differencing-- in KIVA,l KIVA-11 can use a quasi-second-order upwind (QSOU ) scheme for convection. Based on the ideas-of Van her, 22this scheme is monotone and approaches second-order amuracywhem~-onve~%i~zg~o~t?~=~fi}es. Vi%i}e more a~mtira%=thanpm%hd donor cell differencing, QSOU is also more time-consuming, and thus it is included as an option. The number of species and chemical reactions that can be accounted for in KIVA-11 are arbitrary; they are limited only-by ~mmputertime and-storage ccmsideraticms.. The mde distinguishes between-slow reactions-, which proceed Mnetica-lly, and fast re-ac-tionsr which are assumed to be in equilibrium. 23 Chemical rate expressions for the kinetic reactions, which are Arrhenius in form, are evaluated by a partially implicit procedure. Two implicit equation solvers are available to compute chemical equilibria-- a fast algebraic solver for hydrocarbon/air combustion24 and an iterative solver for more general circumstances.25 !Ikvo.modekmre-available to represent the effects of turbulence. The user has the option to use a standard-version-of the k .– c.turbulencemodel,2G modified to include volu- metric expansion -effeets27and spray/turbulence interactions,4 or +~use-a-modified version oi%he subgrid-s~mde(SGS) turbulence model of K117A.1 The S(%3model reduces-to the k —e- mmtel-near watlk -wh~--all ‘tur+iulenc~%q@ seaI&s are too small to be resdied by the- computational-mesh. Kound-ary layer drag and-”wali-”heattransfer are cakuiated””by inatchingto a~mmli~led+mrbulentlaw of thewall. KII~A-11-does-not. -, have a model for the effeets.oftwbulence on the -mean ehemieal reaetion ra-tes~butthe user c-aneasily modify the code to include a mixing-controlled chemistry model.2$-31 E~’apora%ing-liduidspraj!s-are represented byadiscrete-part.icle technique,32 inwhich ea-ch computational particle -represents a number of droplets ofidentiiiaisiie, ve- lb.city, and. %em~erature.L Probability-distributions often govern .tlie assi~ment:of”drop~ ~t:

5 properties at injection or the changes in drop properties at downstream locations. When this is the case, droplet properties are determined by using a Monte Carlo sampling technique. The particles and fluid interact by exchanging mass, momentum, and energy. The momentum exchange is treated by implicit coupling procedures to avoid the prohibitively small timesteps that would otherwise be necessary. Accurate calculation of mass and energy exchange is ensured by automatic reductions in the timestep when the exchange rates become large. Turbulence effects on the droplets are accounted for in one of two ways. When the timestep is smaller than the droplet turbulence correlation time, a fluctuating component is added to the local mean gas velocity when calculating each particle’s mass, momentum, and energy exchange with the gas. 32 When the timestep exceeds the turbulence correlation time, turbulent changes in droplet position and velocity are chosen randomly from analytically derived probability distributions for these changes.33 Droplet collisions and coalescence are accounted for,34 and a new model for droplet aerodynamic breakup has been installed.35 Volume displacement32 and thick spray effects on the exchange rates34 are neglected. Because of improvements to the code’s ease-of-use and versatility, for many applications, all required geometrical specifications, initial conditions, and boundary conditions may be specified using the standard input alone. This is particularly true for internal combustion engine applications. The mesh generation logic allows the computational region to include cupped pistons and domed cylinder heads and to offset these relative to the axis of the cylinder. In addition to two- and three-dimensional Cartesian and cylindrical meshes, the code allows the calculation of the flow in a single “sector” of certain three-dimensional cylindrical configurations in which there is an n-fold symmetry about the axis of the cylinder. This symmetry is often found in engine cylinders with multihole injectors. For initial conditions, one can specify an axisymmetric swirl-velocity field with a Bessel function profile and a specified swirl ratio. Standard boundary conditions and rezone logic allow the mesh to follow the motion of a piston. In response to many users outside the automotive engine design community, a number of other features have been incorporated in KIVA-11. These include gravitational terms, the options to calculate with purely Eulerian or Lagrangian meshes, and inflow and outflow boundaries. The latter are included only for the special case of inflow at the bottom and outflow at the right or top of the mesh, but it is hoped that using these as examples the user can easily modify the code for other inflow/outflow conditions. The user is aided in the task of code modification by the modular structure of the program and a new alphabetized epilogue that allows one to easily find the definitions and uses of FORTRAN variables within the code.

6 II. THE GOVERNING EQUATIONS

In this section we give the equations.ofmoiionfor. t.hefluid.phase, followed .byt,hose Emthe spay cimplets, .a_lld’final!y thelleumky. Cmlciitiolls. ~-~r.~mpactness--ti~s= ar= written in vector notation with bold symbols representing vector and tensor quantities. ‘I%e unit-vec+kn-s-Inthe x-~y~ andz-~r.w-tion~e-dematedby. ii j_iandk.ws~ecti~rel~~: ‘l%e- position vector x is defined by

~ =;&_-+ -J+.-+~k.-, the=veeto?operator? is givemby

.3.. y.= ;-_ +++ ~, &- & W and thdluid velocity veetor=ti=isgiven-by

u = u(x,y,z, t)i+ u(x,y,z, t)j + uky,z,t)k , where t.is time.

A. The Fluid Phase The KIVA-11 equations can be used to solve for both laminar ardturhlent.flows.. The mass, momentlw,and.emer~. equationsf~r. the two e-asesdiffer.primarily.i nthe form and .magnitu(kof thetransport bmeffkients-(i.e., viscosity; thermal ‘cumhxtivity; an&speci~sdifllisitiity); which are much larger in the turbuiimt ~msebecause ofl.he addition a-1 transpcdc-aused by turbulmtfluetuations. In the tu~ulelti-case-the-tr amsportcoel~ici- ~nt~.ared=i%l~ from .a-turb~lentdif fu~ivity that-&pn*on-th-e-tur’n ulent”kinetic energy an&itmlissipation rate; ‘~ecmi.inuii.y. equatibn for species m is

ap.,,r. ~“+ v“. (pmu)=-v. l’wal+’:+’’’~l’

J? I.. :A= w.her.e.pmeiskhema~de~~~.Y_Qf.sn~rie~-f----”” ..->m. ~n.t~.t-o~m~.~a%v .Q .U”u. . . . u -uwuA+=.:3TU.Y_l~~?.&aid ‘H ‘t-il~--iiu”iu‘VW}W= ity. We assume Fick’s Law diffusion with asin~e.fiffti~n.eae~ lcient D., Equations -fcmD- and.source terms due tQ chemistry. PC.~..and .th~~pl%l~~s Willb @V&l i~t~R ~E?Ci(5S “1-iS- the-species of which the spray dropiets are composed, and”ti is the Dirac delta function. By summing-Eq. (1>.over-all speeiGs.-weobitif ithe:tntal-fiuid=d@nsity--equation

7_ de ~+v. (pu)= b’, (2) since mass is conserved in chemical reactions. The momentum equation for the fluid mixture is

a(pu) —+v. (puu)= –&p- AOV(2/3pk)+V-o+~’+pg, (3) at cl’ where p is the fluid pressure. The dimensionless quantity a is used in conjunction with the Pressure Gradient Scaling (PGS) Method.3G This is a method for enhancing computational efficiency in low Mach number flows, where the pressure is nearly uniform. The user may opt not to use the PGS method, in which case a - 1. If the PGS method is used, then a, which varies only in time, is determined in a manner described in Appendix A. In Eq. (3) the quantity AOis zero in laminar calculations and unity when one of the turbulence models is used. The viscous stress tensor is Newtonian in form:

a=p Vu+(Vu)T +AV” U1. (4) [ 1

The first and second coefficients of viscosity, p and L, are defined later. The superscript T denotes the transpose and I is the unit dyadic. F’ is the rate of momentum gain per unit volume due to the spray, to be defined later. The specific body force g is assumed constant. The internal energy equation is

a(pl-1 —+v Q(pul-l=-pv”u+O– AO)O:VU–V”J+AOPC+QC+QS, (5) at

where I is the specific internal energy, exclusive of chemical energy. The heat flux vector J is the sum of contributions due to heat conduction and enthalpy diffusion:

J=–KVl’-PD~h~V(p~/p), (6) m

where 2’ is the fluid temperature and hm the specific enthalpy of species m. The source terms due to chemical heat release Qc and spray interactions Qs will be defined later. When one of the turbulence models are in use (AO = 1), two additional transport equations are solved for the turbulent kinetic energy k and its dissipation rates:

8 (7)

and

(8)

These are standard k – e equations ‘~-with some added terms. The source term (c~~.-~.c~l) V*Uin the-s=eg@ionaccnunts for. length .sde-Am~s.>~h.Ikw_is-v4.asiIy.dila*a$JaR. Source terms involving the quantity Ws arise due to interaction with the spray. Later we . will defhe W-sand@~’~it~@y4eAM~ifimnm. m.] . . 1 mqua-ntltles-cL ~,ctz, ct~,F’r~,and Prt are constants whose vaiues are determined- from experiments and some-theoretical considerations. Standard values of these constants are often used in engine.cahmlatkms, and ‘these are g5ven- imT-abie-Ilbeknv: Avalue @f”cs. equal to 1:50 has ‘been suggested,37 based on the.postulate.oflmgt.hscale conservation in spray/turbulence interactions, and has been found to give good agreement with measure- ments of diesel sprays.4 W-hen -the SGS -turbulaemodel is:use~the.value.dk is ~anstr-ti~~:rti.u-s=tisf~’the inequality

(9)

L~~%~is-an-input-SGS iength scale whose value is typicail~ taken to be 48x, where 8Xis.a representative-comptiatitmai .rmlldimension. Inequality (9) isemforc~xl by integrating Eqs. (7) and (8) in time at all points and then setting c equal to the right-hand side of Eq. (9) atpints-where the inequality is violated; Since. k”?f2/e.iiJmn~ti6naitQ the k-e [email protected] scale;.Eq;.(9) "isa..#~ti*tit+be:t'[email protected]&be-less-$baR.9F eq13*l’L~~~-i

STANDARD VALUES OF k–c TUR33ULENCE MODEL CONSTANTS

9 Near rigid walls this constraint is always satisfied (see the Boundary Conditions section), and thus the standard k-e equations are solved near walls. In regions where the length scale is LSGS, the model reduces to a one-equation SGS model similar to that of KIVA.l The state relations are assumed to be those of an ideal gas mixture. Therefore,

P = ROT I fP#$’~) , (lo) m

1(TI = ~ (pm/p)zm(’rl, (11) m

CP(T-)= ~ (pm/P)Cpm(n , (12) m and

hm (T) = Im(’~ + ROT/Win , (13) where Ro is the universal gas constant; Wm, the molecular weight of species m; lm(T), the specific internal energy of species m; and Cpm, the Specific heat at constant pressure Of species m. The values of hrn(~ and cprn(T) are taken from the JANAF tables.38 The chemical reactions occurring in the system are symbolized by

(14) m m where Xmrepresents one mole of species m and amr and bmr are integra stoichiometric coeffkients for reaction r. The stoichiometric coefficients must satisfy

(15) ~(amr-bm)w m =0, m so that mass is conserved in chemical reactions. Chemical reactions are divided into two classes: those that proceed kinetically and those that are assumed to be in equilibrium. Kinetic reaction r proceeds at a rate d, given by

A,= kf, n (pm/wm; m’ - kbr n (pm/wml”mr. (16) m m

10 Here,the,rewtion.tmdws.a! ,%rand. b!,=-r-neednotequal a~-r and bmr, sa that emplriea-1 reaction orders can be used. The-coefilcients k~~arid k~~are assumed to beo~a- generalized A-rrlleniusrf”ornx

and

(.~7j . .

‘whemE~~ an*E~j- are-activation temperatures. Therties-nf-e@libtiulmr~a-ti-m-s a-reimplicitly determined by Wwcms%rai-nt= conditions

where Kc’(T.., the concentration equilibrium constant, is assumed to be of the form

.K; = EZJL{.A, e.QT.A -!-B;! TA +.C7r+ DrTA +E,T:} ,

-where TA = 271000 K. Withtke .reacticms,rate~ch+ d~er.mined.by-E-qs . :U3)”OP.”(”1”8)7the Clremit%fl”sxraree term in the species continuity equation is given by

-c P m-= ‘m.~ ‘b.ti – ‘J+, , r and the chemical heat relea-sdermin -the energy equaticm is -g-ven by

where Qr is the negative of the heat of”reaction at absolute zero,

(Jr = ~ (Lmr – !h#hprn , m and-( ~hj-”),%is the heat of formation of speciesnmtahmiute m.ro. The transport coefficients in KIVA are taken to be

l.l=(l.o– AO)PVO+ pti, + AOCPk21e ,

A= A3P,

K=:, and

D=~. (23) psc

The diffusivity U. is an input constant, and cp is an empirical constant with a standard value of 0.09. A Sutherland formula is used for ~a~r:

Al#2 —— (24) ‘oir– T+A2’ where A 1and AZ are constants. The constant A3 is taken to be – ~ in calculations of turbulent flow but can be arbitrarily specified in laminar flows. The Prandtl and Schmidt numbers, Pr and Sc, are input constants.

B. The Spray Droplets Solving for the essential dynamics of a spray and its interactions with a gas is an extremely complicated problem. To calculate the mass, momentum, and energy exchange between the spray and the gas, one must account for a distribution of drop sizes, velocities, and temperatures. In many sprays, drop Weber numbers3g are larger than unity, and drop oscillations, distortions, and breakup must be considered. Drop collisions and coalescence have also been found to be important in many engine sprays.34*~0-A2A mathematical formulation that is capable of representing these complex physical processes is the spray equation formulation. 43 In this formulation we solve for a droplet probability distribution function f, and in KIVA-11 f has ten independent variables in addition to time. These are the three droplet position components x, three velocity components v, equilibrium radius r (the radius the droplet would have if it were spherical), temperature T~ (assumed to be uniform within the drop), distortion from sphericity y, and the time rate of change dy/dt = y. We keep track of the fundamental mode of oscillation corresponding to the lowest order

12 . . . . Sphw”ic-ai“Zm-d :;h~~~~~~&~~‘-w.l~fi.~& ~li@~~:-~it~y-~h~-~i ~~iv~-velOCityvector between the droplet and gas. The dimensionless quantity y is proportional to the displacement of the droplet surface from its equilibrium position divided by the droplet radius r. Droplets -bleahl@%mEc@v. ii:j. > lll-?~= The droplet distribution function fis defined in such a way that

,f(xr v, .r, :!d, y, j, t) dv-dl- &rd dy dj is the ~mbabie-mrtir-of -d~opiets-per unitvoiume at position x and time t with vei0citi6s in .t.hehterva] {v, v -Edv)~ radii in the interval (T-$-r--i- ch-)}temperatures in the interval” (.Pdi Ttj. + dTd.), and dis~] at~ent parameters.in the .intermds {Y~Y‘~ dY) an~(.1~.~ ~ d~~). T-wo moments of f have important physical significance. The liquid volume.frmtinm(l, given by

is assumed to be small compared to unity in our equations. The.liq.tidmmcrnsmpic density. p’tt given-by

wherq~ is-t}l~li~ikimie roscopicdensity, c%n-llevetiileiess-’tie-comparable to or iarger than the gas density p because of the large ratio of p~ to p. ‘The density p~ is assumed- mns+ant. The ti-me evoiution of fis obtained by solving a form of the spray equation,

In Eq. (25), the quantities l?, R, $~, and y are the time rates of change, following.an individual drop, of its velocity, radius, temperature; and oscillation velocity y. E-xpres- Sions forthese -willb-=given-lat.er.=‘I%e terms ~co~~ad ‘~~Uare sources dtie to dfoplet coili: sions ana’breakups, and”we now def~ne these. The collision source term ~COl~is given by , Feel{= ;. ( j f(x, VI,rlt7diJY:7j12t) f(x, v2,r2,7’ y j ,t)rr(rl + rz~lvl- vJ -. dj’‘ 2’ I

13 {u (v, r,T~,Y,jI,VI,rl,l“dl,Y1,jI1,V2,r2,TdiY2,~2)

–8(v–vl)6(r –r1)8(Td –Td)6@–~1)6(j-jJ} 1

dvl drl dTd dyldjl dvz drz dTd dyz djz . (26) 1 2

The collision transition probability function a is defined so that odv dr dTd dy dy is the probable number of drops with properties in the implied intervals that result from a collision between a droplet with subscript 1 properties and one with subscript 2 properties. Two types of collisions are accounted for. If the collision impact parameter b is less than a critical value bcr the droplets coalesce, and if b exceeds bcr the droplets maintain their sizes and temperatures but undergo velocity changes. The critical impact parameter bcr is given by

b:, = (rl + r2)2min (1.0, 2.4/Xy)/WeL) ,

f(y) = y3 – 2.4 y2 + 2.7 y ,

y=rJrl where rl < r2’

(27)

The quantity a is the liquid surface tension coefficient, which is assumed to vary linearly between reference value ao at reference temperature To and zero at the fuel species critical temperature Tcr. The precise form for o is

14 where

and

b —bcr r~vl + r~v2 + r; (v2_– v]) r1+r2—b cr V2= r3_+ r3- 12

JiustifYcation for Eqs. (26)-(28) is given in Ref. 34. The breakup source term ~~~is given by

(29)” fbU = ~ f(x, VI, rl, Td , l,jl, t)jl~ (v, r,~d,.y,j, Yl, rl, T~ , jl, x,tl dvldrl dTd djl . 1 1 1

The breakup transition probability function B is defined so that Bdv dr dTd dy dy is the probable number of droplets with prop_@esin the i.mplihdintervzds thatare wacbxxdh.y the ‘breakup ofa dropkt with subsmipt 1 properties. The mwuring-of%q. (29) is-the-follow= ing-: whena droplet’sdistortion yexceeds-’unity, itbreaks-’up into a-distribution of%mller drops given by If. We obtain the totai source to f by multiplyin~the local flux of droplets through. the.~~rface. ~ S ~-by E and ~fi+e~atiflgO_Ve~:th@.~~tiE~-~-E~s ~~. . . .Afterbmakup we assume the dropletrdiifo]lo~va-x-squarddistribution:

1- g(r) = =e.–r’r ,. .{SQ) .% 15 where the Sauter mean radius r32 s given by

=3;= ‘1 (31) ’32 7 1 pdr~ .2 ;+-— 8 a(Td ) ‘1 1

The product droplet velocities also differ from that of the parent droplet by a velocity with magnitude w and with direction randomly distributed in a plane normal to the relative velocity vector between the parent drop and gas. The quantity w is given by

(32)

The precise form for B is

B=g(r)~(Td –Td)8@)6(j)~ NV – (vl + wn)ldn , (33) 1 \ where the integral is over normal directions to the relative velocity vector. Justification for Eqs. (30) - (33) is given in Ref. 35. We now define the functions F, R, Td, and y that determine the trajectories of individual droplets. The droplet acceleration F has contributions due to aerodynamic drag and gravitational force:

3 p \u+u’–v[ F=–— (u+u’–v)cD+g. (34) 8 pd r

The drag coefficient CD is given by

— ~ (1 + l/6 Re~) Red < 1000 CD= d (35)

0.424 Red > 1000 where

2p\u+u’–vlr Red = lltir @) ‘

16 and ~a~ris given by Eq. (24). The gas turbulence velocity u’ is added to the local mean gas velocity when caicui”ati”ng_adropiktk dragand vaporization rate. It is assumed that.each. component .u’ follo.ws.a.Gaussiandiatrihution -with mean square deviation 2/3 -k-.T-bus-we assume

T-hevalue of-u’ k &osammce-wery turbulence-correlation time ttUrband “isotherwise iieid” constant. The droplet correlation timeisgivem hy

/. ~.. &J2 ~ \ t =min -,c — turb ( & ~ & /u+u’–v[ ) ‘ (37)

.-. (pD)tir(T) Y;-–Y1 R=– Shd , (38) 2Pdr 1–Y;

where $lh~ is the S-herwood number for mass-transfer, ‘YI* is the fuei vapor mass fraction at the di-opiet’s surface, Y1 = pi/p, and (pD)air(~ is the fuel-vapor diffusivity in air. The Sherwood number is given by

Sh d = (2.0

Pair(?) Y; – Y1 where SCd_z ~d Bd = . The-surface.massfrm3iomYi* is obtained .frorn ~._y.* pllar(ti 1

[40)

*I17 where Wo is the local average molecular weight of all species exclusive of fuel vapor and pu(!l!’~)is the equilibrium fuel vapor pressure at temperature T~. To obtain Eq. (40), we have assumed that the droplet temperature is uniform and that the partial pressure of fuel vapor at the droplet’s surface equals the equilibrium vapor pressure. For the vapor diffusivity in air we use the empirical correlation

‘2 (pD)tir(T)= ~1~ > where D 1 and D2 are constants. The rate of droplet temperature change is determined by the energy balance equation

43., pd; nr ct7d– pd4nr2RL(Td) = 4rzr2Qd , (41) where Ceis the liquid specific heat, L(T~) is the latent heat of vaporization, and Q~ is the rate of heat conduction to the droplet surface per unit area. Equation (36) is a statement that the energy conducted to the droplet either heats up the droplet or supplies heat for vaporization. The heat conduction rate Q~ is given by the Ranz-Marshall correlation:45

Ktir@’) (T – Td) Qd= Zr Nud , (42) where

tn(l + Bd) ‘Ud = ‘2”0 + O“6R’H ~ ‘ d

lldr(ficp(a Prd = K@) ‘

cP is the local specific heat at constant pressure and at temperature ~ = (T + 2T~)/3, and K1 and K2 are constants.

18 Consistent with the approximation that the liquid density is constant; we also assume its internal energy 1/.is a function of temperature.alone.. Thust.heliquid-enthalpy- ‘will.h&Jea.small.pressdre”depe3deflce;

he(~d,p) = Ze(Td) + p/Pal. (43)

Si.nee-tlheki+tent~hea%of ‘~aporhtbn L is tile energy reqyired- to convmti.= =un-itmassof. liquid to vapor at constant pressure equai “tothe equilibritim vapor pressure, .the.liquid and-vapor. enthalpies and internal ene-rgies-and L a-rerela%d by-

(44) L(Td) = hl(rd) – ~#’d,P#’d)) = I1(Td) + RTd/W1 – z$Td) – Pu(~d)/Pd .

The equation for the acceleration of the droplet distortion parameter is

8 a(T~) 5P$T~) .. 2 p (U+ U’– V)2 Y=3pd.— --Y– 5’, (45) ~2 pd?-s pdrz

-WhEW~e(Td) isrthe-viscosity ofthe-iiquid. Equation (45), which is based ‘on the analogy between an oscillating droplet and-a spring=m~~y.4m,4G.i=tha.equation..of.a-furcedj damped harmonic oscillator. The external force is supplied by the gas aerodynamic forces on the droplet. The restoring .force is supplied by.surface tension forces. _Damping issup. pl iedhy Ii@& viscnsity. ..kietied.diwwssion.=of Eq. (.* ) .m=~~.~a f~t~~>-.-u~l,<.~... ..=&...... ““.2s.. . . ~,~.~a~~new in .~-positi~~.~o@v~ ~h~ e~~~~ng~furrctionsp, FS;QS;aM”d””’WS:(i%ese are obtained- ‘bysmlm in@he_raksmf.cliange of.mass,. mmnemtum$ and energy of--a-lldio~kts at position x and time t.34 Thus one obtains

F%=-r! f pd(4/3 rrr3F’ + 4nr2R v) dv drdTd dy dj , (46)

and=

*s=_i-j fp.d.arm3 $:. ii dv.dr d.Td .~zy.a’j.-, ~ ~~. where F’ =F–g. Physically, ~sisthen egativeo ftheratia twhicht heturbulent eddies are doing work in dispersing the spray droplets. Since u’ follows the Gaussian distribution (36) it can be shown that W <0, and thus this term always depletes turbulent kinetic energy. c. Boundary Conditions In this section we give the physical boundary conditions that are available as standard options in the ICi_VA-11code. There are also numerical boundary conditions used by the program in conjunction with inflow and outflow boundaries. Numerical boundary conditions are extra conditions that are not required by the equations themselves but that have been found through experience to be necessary in implementing computational boundaries in fluid flow codes. 14 To understand these numerical conditions requires some familiarity with the numerical solution procedure in KIVA-H, and for this reason we defer discussion of inflow and outflow boundaries until Sec. IV.K of this report. In addition to inflow and outflow boundaries, two types of physical boundaries are available in KIVA-H -- rigid walls and periodic boundaries. There are, in turn, several types of rigid walls depending on velocity and temperature boundary conditions. The velocity boundary conditions on rigid walls can be free slip, no slip, or turbulent law-of- the-wall. Temperature boundary condition options are adiabatic walls and fixed temperature walls. In engine calculations one ordinarily uses turbulent law-of-the-wall velocity conditions with fixed temperature walls. We now give in detail the rigid wall boundary conditions for the gas-phase equations. Velocity boundary conditions on rigid walls are introduced either by imposing the value of the velocity on walls or the value of the wall stress OW= an, where n is the unit normal to the wall. On no slip walls, the gas velocity is set equal to the wall velocity:

k, (47) u = ‘wCdl

where the wall is assumed to be moving with speed Wwazzin the z-direction. The wall stress is then determined implicitly through Eq. (3). On free-slip and turbulent law-of- the-wall boundaries the normal gas velocity is set equal to the normal wall velocity,

(48) ‘“n=wud k.n,

and the two tangential components of OWare explicitly specified. For free-slip walls the tangential components of Oware zero. For turbulent law-of-the-wall conditions the tangential components are determined by matching to a logarithmic profile:

20 (49)

where-

(50)

where v = u — WWaZ/iL In Eqs. (49) and (50) it is assumed that y is small enough to be in the logarithmic region -m%lmlamina~sublayer~@on-of the turbulentbmmdary laye~n The Reyrrolds nu.m-br.-~e dd-in~.the.-b~undary. -hetwthmhtwowo mgibns.- The rmnstants ~, c

/ K=-vc:(c –Ce )17: ?2 -1

and

~or, ~~~~ly ~~~~~~~dV~lU~S Of ~h~k.+. canstantsi B =S.s j and .C~tiG().IJ5 ; W~obta-in K = .4W7 am-l?c =114. A der=vaticmof-i3qs~ (49) -(Y1-)isgive-rTi~z-A-~zl*x-B. ‘Temperature boundary conditions on ri~d”wails are introduced”by specifying either the wall temperature or the wall heat flux Jw = – kvZ’”n. For adiabatic walls, we set Jw equal ‘mzeru. Futilxed-tel~erdtum--walls that-am--alsoeitilerfme-sli~or-no-sfiip; the--wall temperature is prescribed; and “J~ is determined ”irnpiicity from Eq. (5). For fixed ”temperature walls using the turbulent law-of-the-wall condition, Jw is determined from the modified Reynolds analogy formula

where.TW isthe.wdl.tempratme.m~rt isthe.P~andtlnumber. of.the.laminar. fluirL .-A. derivation ofEq. (52) is-givainAppendix-B, ~-1 In addition to the wall heat loss, there is a source to the internal energy due to frictional heating. Frictional heating occurs whenever turbulent law-of-the-wall velocity conditions are used and has the form

= P(U*)2U , where fWis the heating rate per unit area of wall. In calculations of turbulent flow, boundary conditions are also needed for the turbulent kinetic energy k and its dissipation rate e. These are taken to be

Vk. n=O and

where k and c are evaluated a distance y from the wall and

c k P c Pc = [ Pr$cc2 -ccl) 1“

Periodic boundaries are only used in KIVA-11 when the flow field is assumed to have an N-fold periodicity about the z-axis. When this assumption is used, the computational region is composed of points in the pie-shaped sector O~ @~ 2dN, where 6 satisfies cm 6 a x/_ and sin 0 = y/-. The periodic bolmdaries are those for which (3= Oand (1 z 2dN. The conditions imposed on these boundaries can be inferred from the assumed N-fold periodicity. For a scalar quantity q the requirement is that q(r, (3,z) = q(r, 0 + 2rdN, z), where r = g. For a vector v the requirement is that v(r, (3 + 2rt/N, z) = J1.v(r, (3, z), where R is the rotation matrix corresponding to the angle 211/iV. Boundary conditions are also needed for the spray equations, and we indicate here what these are for a spray injector and for a rigid wall. When a spray droplet impinges on a rigid wall, we set its velocity equal to the wall velocity, and for purposes of calculating heat and mass transfer with the gas, we set Re[f = Oin Eqs. (39) and (42). There is no heat transfer between the droplet and wall. This is a provisional model for spray/wall interac-

22 tmns-wmt wil’l=unabufitedllyciian-ge as-tii3=iin@TtipTm:titim-re~WFteS-IfiOm at%-entiefi= firam.engine.researchers. Armther typed physical bmmdary for the spray equation.is thespray.injectm.. This is-a.point in.space at w-MA w.e.spedy dmpkt-mass. flow rate and a distribution of drop- letsizes, velocities, temperatures, and oscillation parameters.. An arbitrary number of spray injixtors.may be used’in a singlefiTA-fic%tcuiatio~li T-lE-mass fimv rate =foreaeb

ifijiix%or-ii ~mnstant”between times-”Tl-.~nland TZ inj and is-=zrmothemvi~e: ‘W~~“t.~P~llf~il= jeetor- size Fig; 1. The directionof the spray axis can be arbitrarily speei - fi-ticlnee~-not .coincide with the z=axis-ofit,ke-flow field=cootiinate-syste~n: ‘Thepolar- angle @of the droplet veloeity mla-tive +Wthe spray axis is distributed unifor.mly.in the interval [O.in,o.~~~l,and the azimuthal angle 6 is distributed uniformly in the interval [0, 2n]. The temperatures of all injected drops have the same value T~inj. If the droplet breaklqmod.e~kin use, then all injected- dro@mve-;~ =0 -aml<~= ~~nj, ‘where ~~njis the produetof aninputdimmsionl~ s-=plitude and dimensional frequency. .A-.moredetailed dese~lption of-the dropletirrjeetion [email protected] Program section of *his-repo A.

F&.. i: S@ieri6ai’cocwainate system-useftu define the distribut~cm ofdireetiorumf= droplet-velocities at a spray. injector. ~~ III. THE NUMERICAL SCHEME

KVA-11 solves finite-difference approximations to the governing equations of Sec. II. The equations are discretized both in space and time. Before specifying the numerical scheme in detail, we discuss some of its general features and the principal ways in which it differs from the KIVA numerical scheme.

A. Temporal Differencing The temporal differencing is performed with respect to a sequence of discrete times @(~ = O, 1,2, ...). The time interval Atn = tn+ 1 – tnis the timestep, and the integer n is the cycle number. The latter is displayed as a superscript, so that Qn denotes the difference approximation to the quantity Q at time tn. When At appears without a superscript, Atn is understood. The difference approximation to the derivative ~Q/iltis the first-order expression (Qn+ 1 — Qn)/At. It is in its temporal difference scheme that KIVA-D differs most from KIVA. Just as in KIVA, a cycle is performed in three stages, or phases, but the terms difference in each phase and their temporal differencing have changed considerably. Phases A and B together constitute a Lagrangian calculation in which computational cells move with the fluid. Phase A is a calculation of spray droplet collision and oscillatiordbreakup terms and mass and energy source terms due to the chemistry and spray. Phase B calculates in a coupled, implicit fashion the acoustic mode terms (namely the pressure gradient in the momentum equation and velocity dilatation terms in mass and energy equations), the spray momentum source term, and the terms due to diffusion of mass, momentum, and energy. Phase B also calculates the remaining source terms in the turbulence equations. In Phase C, the flow field is frozen and rezoned or remapped onto a new computational mesh. For the detailed description of each phase that is given later, it is convenient to define intermediate quantities that have been partially but not fully updated. Such quantities are identified by superscripts A and B. Thus, for example, QA is the computed value of Q at the end of Phase A. (Superscript C is not needed because it is equivalent to superscript n + 1.)

B. Spatial Differencing The spatial differencing is based on the ALE method, 17’18which in three dimensions uses a mesh made up of arbitrary hexahedrons. Spatial difference approximations are constructed by the control-volume or integral-balance approach, 17which largely preserves the local conservation properties of the differential equations. The spatial region of interest is subdivided into a number of small cells or zones, the corners of which are the vertices. Together, the cells constitute the mesh with respect to which spatial differences are formed. The vertices need not be stationary, but

24 may move in an arbitrarily Prescri’bed’manner. “TItiiS-ca_pabilityincludes-th&La-grangJan and Eulerian descriptions-as special cases. In the gener.aIcase, the cells are asy.rnmetri~ cal; a typical cell is shown in Fig. 2. The vertices are conventionally numbered-as shown. The Pdk areindexed by integers (i,j, k), which may tie regarded as coordinates-in- klgica..spac% T!~ndicM~i,j, .k).also label the vertices, with the understanding that ve.rkx (i, j,. k)is-vert.ex 4 for cell (i, j, k). The CartesiticQordi nates of”vertex (i,j, k) a~e-

(X~~,y~k, Z~k), Whkh in genera~”depen~ on”~h~-ti~me”t. Thus the position .vec-+arto vertex- [i:j, k).is

X.. = x.. i-+y. j+ z,, k (54) LJk lJ k LJk lJ k

The “center” of cell (i, j, k) is defined as the point with coordinates

-— 1. .: z:, = — (55) LJk g “+1‘. ‘ -----

,. 5 I 1

j .— k i

Fig; 2: Typical ikite-differencecell, where (xa, ya, Za) are the coordinates of vertex a of cell (i, j, k). In general, the point (xcijL

YCW, ZWJ is not the center of mass or volume of cell (i,.j, k). It is convenient to define auxiliary cells centered about the vertices. These cells are called moment urn cells as their main use is in differencing the momentum equations. Momentum cell (i,j, k) is centered about vertex (i,j, k). In contrast to regular cells, which have six faces, momentum cells have twenty-four faces, each of which is comparable in size to one-fourth of a regular cell face. Three of these twenty-four faces lie within each of the eight regular cells which share common volume with the momentum cells. The portion of momentum cell (i,j, k) lying within regular cell (z,j, k) is shown in Fig. 3. The points of intersection of the momentum cell faces with the regular cell edges are defined as the midpoints of the regular cell edges. The points of intersection of the momentum cell edges with the regular cell faces are then defined implicitly by the requirement that the regular cell face be partitioned into four subfaces of equal area by the momentum ceil faces. The corners of the momentum cells are then implicitly defined by the requirement that the overlap volume between a regular cell and a momentum cell centered at one of its corners be one-eighth of the regular cell volume. In general, the momentum cell corners do not coincide with the cell centers defined by Eq. (55). The momentum cell corners and the intersection points of momentum cell edges with regular cell faces are not actually solved for as they are not needed. The location of velocities at cell vertices in the ALE method is convenient because no interpolation is required when determining vertex motion in the Lagrangian phase of the calculation, but it has a major drawback. This is that ALE method solutions are notori- ously susceptible to parasitic modes in the velocity field. A major reason for this is that pressure waves tend to propagate along cell diagonals rather than via adjacent cells.t7J8 A “checkerboarding” effect is thereby created in the pressure field, with associated irregularities in the velocity field that are usually suppressed by the introduction of a numerical damping called node coupling. 16’17In a major improvement to the ALE method, we have alleviated the susceptibility to parasitic modes by the introduction of velocities centered on cell faces.2 Vertex velocities are retained, and momentum associated with the vertices is conserved, but normal velocity components on cell faces are used to compute cell volume changes in Phase B and fluxing volumes in Phase C. The resulting scheme greatly reduces the need for node coupling, and many problems can be run with no node coupling at all. Accelerations of the cell-face velocities due to pressure gradients are calculated by constructing momentum control volumes centered about the cell faces. Like the momentum cells the cell-face control volumes have twenty-four faces. Referring to Figs. 2 and 3, the cell-face control volume for the left face of cell (z,j, k) is composed of those portions of

26 <. / 1. [ i /’ .—I [ / I / ,- .– (-

Mg; 3. T-he-portioRofi mo.mefitum”cell {i; j; k) lying within regular cell (z,j, k]. The three nmn-rentum-cdl faces l.yi w within the re=alar cell are shaded, Each momentum cell has twenty- four such faces in all.

the momentum cells of vertices 3,4,7, and 8 that lie in regular cells (z,.j, k) and (i – I,j, k). Control volumes associated.with.the-other-cell.faces are defined analogously. The volume of any mmnentum amtrol volume may be calculatedomce-the-volumes of” the main computational cells are known. The volume of cell (i,.i, ~) is denoted by Vijk ad is calcuiated”by the following formula:4g”

c: = – (Y2Z3-t-y@4 – y2z5 – Y’$6 – 3’3.Z2+ Y3Z4 – 3’4~~-–Y4%-+Y&

+ 3’4ZS + Y5Z2. – 3’5!4 -. + ~.$fi – Y.&~ + Y&~ – Y&~ – Y8~4_+ Y8ZJ > C6 = (Y1Z2– Y1Z5 – yfl + ’223 – Y.2Z5+ Y~7 – Y3Z2+ y327 + yfl

+ Y5Z2 – Y&.7 – y5zg – y~z – Y#3 + y#5 + Y#/3 + y<5 – Y~z7) ,

C7 = b&3 – 3“2zf5– Y3Z2 + y#4 — Y3Z6 + ’328 – Y4Z3 + y4z~ + Y5zf3

– y5zfj+ Y&2 + Y&3 – Y#5 – Y&8 – Y8Z3 – Y8z4 + Y&5 + YBz6) ,

C8 = – (y124 – y1z5 – y3z4 + y3z7 – y4z1 + y4z3 – y4z5 + y4z7 + Yfl

(57) + Y<4 – Y&~ – Y5Z7 + 3’&5 – Y(F7 – yf3 – Yef4 + Y#5 + y’f@ J and the summation extends over the eight corners of cell (i,.j, k). It is also necessary to know the x-, y-, and z-projections of the surface areas of the cell faces. Since each face is common to two cells, there are three independent faces per cell. These are conventionally taken to be the left, front, and bottom faces of the cell as viewed from the perspective of Fig. 2. These faces are shown in Fig. 4. The conventional direction of their vector area elements is outward from the cell, as shown in Fig. 4. The area projections of these faces are calculated as follows:49

AeX = –0.5[(y3 –Y4)(Z8– 24)– (Y8–Y4)(Z3 – 24)+ tY8-Y7)(z3– 27)– b’3‘y7)(zj3– ‘7)1>

Aty= +0.5 [(x3 – Z~(Z8– Z~ – (X8– X4)(Z3– Z4)+ (X8– ~7)(z3– Z7)– (X3– ~7)(z8– 27)!,

A~== -0.5[(x3- zJ@8-yJ -(z8-xJ@3- YJ+(X8- X7)@3-Y; -(X3 -X7) b8-Y/1 ,

Af== +0.5 [(yl –y4)(ZB– Z4)– (y8-y4)(Zl –Z4) + (y8-y5)(zl-z5) – (y, –Y5)(z8–z5)] ,

28 X4)(Z8 – 24) – (X8 – %C4)(Zl – 24) + (Z8 – X5)(Zl– 25) – (Zl – X5)(Z8 – 25)] , %Y = –o:5T(xl–

A%_= +0.5 [(ZQ – 24) (%1 – X4) - (Zl - 24)(X3 – X4) + (Zi – 22).(X3– X2) – (23 – 22) (X1 — .X.-J] , .

A (58) bz = – 0.5 [(.Y3 – Y*) (xl – X4! – (Y, – 3’4!@ –-:4? +“q – 3’2) (xa_– Q – b~ .– Y2) (~~ – X2)1 , where subscripts f, L and b refer respectively to the.left,frnnt, and.b~ttom.faw~.. W-hen- neede~.thesuhscripts r, d, and t will be used fa refer respectively’~ the right, back; md top faces-of a-ucomputational cell. T-hese-area-projections enable us to define area vectors A-aassociated with the faces a o~eacli regular cell; 13is convenient to associate Aa with both the face and the cell; so tha-t.~ may bemmsidered ‘toalways poinhutward-from the eellundermnsideratio~~ Thus; fora-g-iven face, the sign Of”A-adepends on the side of the face from which it is viewed: Gonsider, for example, the left face of cell (z,j, k), as shown in Fig. 4, and compute AtX, Aey, and Atz by Eq. (58). From the point of view of cell (i,j, k), the vector Aa for this face is A~Xi + A@_+ A~Zk. But.of.cell.(i.– I,j, M, the.veetor. .4mfor this face -is-– A~Xi – A~Yj- Atzk.. Similar considerations applyto right, front, back, bottom, and top faces. Similarly, the outward area vector associated .withface.cmf.ti-par.titular mmnentun- cell [email protected]!a, .ancLthe cmtward area vector offae+w of’ a-~dl-face ~controlvo]umeis

I?ig. 4; Cell”faces associated-with cell (z,j,k).

2 ylJ denoted by A“a. We sha not write out explicit expressions for the components or projections of these momentum cell-face area vectors, as it will always be possible to eliminate them in favor of the Aa as discussed below. In the finite-difference approximations of KIVA, velocities are fundamentally located at the vertices, so that

(59) u. (Xtik,Yijk,Zijk lJk = u ).

Thermodynamic quantities are located at cell centers:

(60) where Q = p, p, T, 1, or pm, as well as k and c. Quantities needed at points where they are not fundamentally located are obtained by averaging neighboring values. Spatial differences are usually performed by integrating the differential term in question over the volume of a typical cell (or momentum cell). Volume integrals of gradient or divergence terms are usually converted into surface area integrals using the divergence theorem. The volume integral of a time derivative maybe related to the derivative of the integral by means of Reynolds’ transport theorem. 50 Volume and surface area integrals are usually performed under the assumption that the integrands are uniform within cells or on cell faces. Thus area integrals over surfaces of cells become sums over cell faces (or subfaces):

.A (61) ~’.dA+~F au” I a

When differencing diffusion terms for cell-centered quantity Q, it is necessary to evaluate (vQ)a” Aa. Referring to Fig. 5, this quantity is approximated as follows. The points xt and x, are the centers of the cells on either side of face a, and xt, xh, XL and x~ are the centers of the four edges bounding face a. We first solve for coefficients aer, ath, and afd such that

atr(xf — Xr) + atb(xt – Xb) + afd(xf – Xd) = (62) Aa

Note that since the mesh may be nonorthogonal, the vector x~ – xr need not be parallel to Aa, and thus ath and af~ maybe nonzero. The finite-difference approximation to (vQ)a” A(, is obtained by dotting both sides of Eq. (62) with (VQ)a and ignoring terms of second and higher order in the cell dimensions: 30 h-Eq.(63) “Q.is.t.hesiinpk awx-ag$ OftLie vdms.ofQ>Mlie fbur-c+lI&w*reuHdiRg-.%4l- edge “t,” and Q~, Q~ and Q~ are defined analogously. .4reainte~als~\’er.m ommtum cell faces are ordinarily Converted into area integrals-Oveme@a~-mil ‘fiims-byt.he ftillbwifig procedure: Eet Q-’be a quantity that is uni t6m-witfiiil-regmitir-cei13, and’considix the volume of overlap between regular cell”(i; Jlk) and the momentum cell associated with one of its vertices. Three .faces of.this.overlap vo~ume-(call them a,b,c) are faces of-the momentum call “in question, with outward ‘area vectors A’a, while the other three (call them cl,e,fl are surfaces of regular cell (i, j,k), with outward area vectors ~ Aa. But .the.divergence .theorernshows that the inte=~al 1 dA over- the entire surface of this overlap volume is-zero, so.that.

Thus t.he--inte-grd -i-QdAzover the- three-momentum cell “faces in question my be repre-- sented by

so that the area veetors A’-a never need to be expll-cit3y eva-iuated’. A similar procedure is used “toexpress the outward-normal-areas A’7aof faces of the cell-face control volumes in terms of the regular cell face areas Aa. The mass of cell (i,j,k) is denoted by M~h and is given by

Fig. 5. Tine six points used”to define the gradient of cell-centered quantity Q_on cell face a. (66) ‘ij~ = PLJ~vQ~ -

The mass of momentum cell (i, j,k) is given by

M;k = : (Mijk+ Mi_l j,k +M, +M. . +M, . l–l,j-l,k l, J- I,k L,j,k– 1

+M +M, +M, . (67) t–1,j,k–1 t-1,j–l,k–1 z,j-1,k–1 ).

The mass of left cell-face control volume of cell (i, j,k) is given by

M; = : (Mtjk + Mi_l ~,k) , (68) ● and the mass of other cell-face control volumes are defined analogously. c. Stochastic Particle Technique A very efficient and accurate method for solving for the spray dynamics is based on the ideas of the Monte Carlo method and of discrete particle methods. In discrete particle methods, the continuous distribution fis approximated by a discrete distribution ~’:

f’= ~ NP6(x-xJi5(v -vJ6(r- rJi5(Td -Td)i5(y-yJ6(j-9J. (69) ~=1 P

Each particle p is composed of a number of droplets NP having equal location XP,velocity VP,size rP, temperature Z’~P,and oscillation parameters yP and yP. Particle and droplet trajectories coincide (thus, for example, dx~dt = VPand dv~dt = FP), and the particles exchange mass, momentum, and energy with the gas in the computational cells in which they are located. The finite-difference approximations to the ordinary differential equations for the particle trajectories and for the exchange rate functions of Eq. (46) are given in Appendix C. Our method is a Monte Carlo method in the sense that we sample randomly from assumed probability distributions that govern droplet properties at injection and droplet behavior subsequent to injection. We now show how this random sampling is done. Let us assume that we are given the distribution function fix) corresponding to the random variable x (xl ——< x < x2). The distribution function is defined by dN = fix) dx; i.e., this is the number of droplets in the interval dx about the value x. Let us define the random variable

32 [x y = I /’(x’)&’ , JX 1 and we note that dN = dy. Hence,the number of droplets is uniformly distributed with mspect-’m--thvatiabieyey. Commonly one has availabk randommum-heqymrators with a uniform dtiribtiiti~n.the.range.from=v+va &aone. We therefm~=arnp]e fromthidist=~ bution, scale by

to obtain y, and then invert to obtain x, which then will be distributed according taflz). Depending on the form of the distribution function, the integral and its inversion maybe performed analytically or, failing that, by a numerical method. This sampling procedure iS-USed~tithe-~O@A.~njW$l~n.~QICUlati.On; ‘whie-h-is-4aswi’u@-in-.@pmdix-D; in-the thmp= lel~ollition.r~lculation, which isdesc-ribed in ..Appendix-E; in the dmpletbmaku~calcula-- ticm; de*+#ned=in=A-~ntix=F; ‘a~~d-when-a-newvalue oftiie gas turbulence velOcity U’P must=bechosen. in the latter case, since each componentof ‘u’Pfollows a Gaussian distribution G(u’) and since

(x erf(x/# 4q13 ) = 2 ! G (u’) du’ , (70) Jlrj it follows that we must invert the error function erf. This is done by storing values of the inverse error function erf– 1 (y) evdutitiint~als~f.~.~~ .frewly =0 .%3 ~ = 1.90, TJe value of er~– 1 (1-.0) is -talzen+Abe* -.9 0: ~vra~u~-oferf-~ @at~“ intermediate-vaflues-ofy are found by linear-interpolation-. If-y is-a-random number between 0.0 and”i-.O-,theu Zy — 1 is a random number between – l. O-and-l.O whose magnitude and sign determine the magnitude and sign of u’:

(71) u! = ti4q/3 sign (’2Y–– l)e#–’(12y –-14) .

~f.~~r~e, a.ne\v-u!P-is4nly -s@pl&-once eve~-umfielatiOfi-til~trUr~-.[See-Eq-= (37)11. W-hen the turbulenemmelation timetlU,~ Ifl=spraypatiicie-isless-than the canpu-- tational timestep At, the droplet equations cannot be integrated directly since the particle ‘

33 droplet experiences a linear drag law. When At s ttUrh,we also set U’P = Owhen differencing the droplet equations. Appendix G discusses briefly how the distributions of turbulent displacements are obtained, and a detailed derivation maybe found in Ref. 33.

D. State Relations The quantities Im(7’) are obtained from the JANAF tables38 and are stored in tabular form at intervals of 100 K. A simple linear interpolation is used to determine the lm(T) at temperatures within the range of the tables. The quantities cUm(T)are simply approximated by differences between adjacent tabular values of Im(Z’), divided by 100 K. The temperature 2’ determines the internal energy I via Eq. (11). Conversely, I determines T via the inverse of this relation. This inversion is performed by a simple linear search algorithm which takes advantage of the fact that (tW~T)Pmis always positive. Let Tn = 100 n (n = O, 1,2, ...). and choose the initial value of n so that Tn is a reasonable esti- mate of the correct T. One then evaluates In = I(TU) from Eq. (11). If In s 1, n is reduced, and if In + 1 <1, n is increased. The search proceeds in this way until I is bracketed by In and In+ 1; T is then evaluated by linear interpolation between Tn and Tn+ 1. The liquid fuel internal energy 18(T), the liquid viscosity pe(T), and the equilibrium vapor pressure pu(T) are also stored in tabular form. The liquid latent heat L(T) is first stored at intervals of 100 K, and then the values of I~(T) are calculated at intervals of 100 K from Eq. (44) and known values of L(T), pU(T), and ll(T). Because the vapor pressure pUand liquid viscosity pt vary rapidly with temperature, their values are stored at intervals of 10 K up to the fuel critical temperature Tcr~t. Latent heat, vapor pressure, and liquid viscosity tables for many fuels can be found in Refs. 51-53.

E. Lagrangian Phase Difference Equations With the above background, we are now in a position to specify the KIVA difference equations. It is convenient to give these first for the Lagrangian phase and then for Phase C or the rezone phase. In the equations of this section, we use implicit methods to difference the terms associated with acoustic pressure wave propagation and diffusion of mass, momentum, and energy. In the next section we tell how these coupled implicit equations are solved. 1. Mass Density Equations. The Lagrangian phase difference approximation to Eq. (1) is

(P&k V:k – (P& V;k At

(72)

34 The mass fractions are related to the densities by.

(’73) where-x = ,nj A, or B: The Phase Amass densities will b--k~lned-shortly=. A-nimportant feature of Eq. (72) is the use of variable implicitness parameter $D in differencing the diffusion term. Parameter @D varies in space and time and is defined at cell centers. Its value, which lies between zero and one, depends on the local diffusion Courant number

where A-xis a measure of”the ceil ”siie. ‘WIien Cd is small compared-to unity, @~ is zero and a fully expiicit difference approximation is used. When Cd is large compared to unity, _@~ is c-dose--to--unity-and-animp_licit formulation is used. ‘The exact expression-d’tir.~.~,-which is chosen to ensure. numerical stability, is ~~venin .4ppendix-H. The PhaseA density ofspeeies n-vinehdes mntributioms=dm.e-tnchemistry. and spray ~TJ~~d~~~~~~:

The c-hemistry source-term (p~c)ti~... . _is given by E-q,(20) -with-hr replaced by (ti.~:4)ti~.The. irrte-gration methodfor ‘kinetic. mactiim rates hr,is described=~nAppendix 1, and-thatfbr equilibrium reactions is described in AppendixJ, Thdlnite-differenm approximatiomto thespray mass sxmree terrr-~tik‘-is-@ven-in-Appendix C. Summing Eq. (72) “over ail ‘species m gives the following Lagrangian phase difference approximatitm to the mass dimsity equation {2):

(75)

This shows the total gas_mmsin~ell.cAanges cmly-due.to the spray smmce. %milarly slumming. (3’4Joverall speei~-and cmnparlng with (75) shows t!rmt=Phasw-B--cellmasses are kinown after-PhaseA:

(76)

35 Equation (76) can be combined with (72) - (74) to obtain

(77)

Equation (77) is solved in Phase B. 2. Momentum Equation. The Lagrangian phase difference approximation to the momentum equation (3) is the following:

(78)

– ~ @~~ + S;ku;k) + g(M’)~.k – (M’):.k (ANC)ij~/A~.

The index ~ refers to the faces of momentum control volume (z,j,k), whose normal area vectors at time t~ are (A’)~ ‘. The pressure p, turbulent kinetic energy k, and viscous stress tensor o are regarded as being uniform within the regular cell in which face ~ lies. Thus the A’ can be eliminated in terms of regular cell face areas A as described in Sec. B above. This makes it convenient to evaluate momentum changes due to surface stresses by sweeping over cells rather than vertices in the computer program. In differencing the pressure gradient term, variable implicitness parameter @p is used. Parameter @Pplays a role for the acoustic mode terms analogous to that of @D for the diffusion terms. Appendix H gives the exact expression for @p, which depends Onthe sound speed Courant number

where c is the isentropic speed of sound. The PGS parameter an, which is used to artificially raise the Mach number in far subsonic flows,3Gdepends on time but not on space. The formula used to determine an is given in Appendix A. The viscous stress tensor a is a weighted average of the stresses based on time level n velocities and those based on the Phase B velocities. This weighting employs the same

36 xv~ri~b~~ irnp~i~-it~~SS-p~r~~er +~ thti S--used=foPthe nmss-diffusicm terms: ~~e- VZdUe d -.=..—. G depends-on-velocity ~adknts- whose -aMerence- approximation&m~@ Yen~n-Appendix-_K. T&quantities .Rlti~.and.S’~~-are associated with the-implieit~wtiplingo f-the vain= putdgas-and drop vekwities. one ean-~nteWre@’ ~-ka~-a~-a-ddedmass that arises “because forces on momentum cell”( i,j,k”) must accelerate the droplets in ad~ltion to the gas in tliat:cdl: Tllie appearance ofS’~~-in (78) “is-rekted “totbweiljknown h.vering of the scmrrcl speed -intwo-phase flmvs.q~ The evaluationof-i%~~u.jk and S~~~_is. described in A-ppenciix C-I A-lthougE we have greatly retie~thea=~ptihiliiy-of..~-.s~lutionsntoto parasitic veioci ty modes .orail,ernaknock uncmnp~ imsome .pdile.ms the veloci=tyfield m~y de- velop persistent alternative-vertex irregularities of small amplitude, Such irregularities.< can usually. be.eliminated_by usingt,healternate node coupler deseribed in -.Appendix L. The effects of the alternate node coupler me.r.eWe~nted.by. the.term .(ANC)~~, which is- the-sum-over-atl re-gwlar cells surrounding vertex (i,j,k”) of the terms 5u~kanC”gitienifi Eq. (E-4), The Phase A vertex velocities include changes due to the spray momentum source, grayitationai .accekation, and the aiternate ncik Coupl-&:

(79)

1 .~_a&:&tom~=~J~ ~ll-~& ~. u.e ~] c[e~t~a:. ti~m-.~o-wriex. veioe.ities jin Ph as&3 ‘we use face-centered normal velocities. We compute accelerations of these velocities due to th~$hermod’ynarnicami tiirbulence presswresp anti-%pk;and the-resnitingftie.cemi.~ed: nQrmd-velQcitiesNe.thaq -used.te.~ekukte the .Lagrangifm phase cdl w lwrwichanges. This procedure has been found to reduce dramatically the susceptibility of computed solutions to alternate node uncoupling. 1~17The reasons. for thisimprmememt.are diiwussed in. Refs. 47 and 48. Wenow describe-how the fam-centered-velocitie~ -am-computed. Rather than deaiing directiy withface-centered normai “velocities, it is more convenient to introduce a factor oftlie.cell-fice.are~. TAusthe.variables we use are

(tif%)a= “d- =-A-- us

An equation for u.” Ammybdexived.as follows. Consider the momentum balaned’or ccm- tro~ ~~a]umeT,I$ha-t.mov~-ivi th the-fluid:

; [ puf.iu=F, (81) J v U3‘27’ where F’is the sum of all forces on V. By dotting this equation with area element A that is moving with the fluid, we obtain

D DA pu. Adu=F. A+—. pU du . (82) z I Dt \ v v

In curved meshes, the last term in this equation gives rise to Coriolis and centrifugal force terms. Equation (82) is difference in the following manner.. We initialize (uA)a using vertex velocities ut that differ from the Phase B velocities only in that terms in (78) are omitted that are associated with the thermodynamic and turbulence pressures p and ~ pk:

u!. —u!. (83) [(M’):h + S;jk] ‘JkAt ‘Jk = – ~ ~ [@/B + (1 – @) Pnl$A’)~– A. ~ 3 P#$M’)~ . (an)2 ~ B

Alternatively, ut differs from UAby the addition of the viscous term contributions:

Ut —u! (84) [(M’):k + S;.k] ‘JkAt ‘Jk = >7 [~~~ (UB) + (1 - *D) o (Un)lp.(A’); . 7

The quantities (uA)a are initialized by

(85) (uA):=Hu: +u; +u:+u; )”A:, where a, b, c, and d label the vertices that form the four corners of cell face a. This label- ing convention will be understood in what follows. The finite-difference approximation to (82) is then given by

(uA): – (u/i)t [(m: + s:] At “ = – ~ {[@#B+(1- 4JpnV(an)2+ A03pAkA}Yya(A’’)n.A Y

At – A; U:+u; +u:+u; +a. [(M”): + S:] (86) Al 4

The indices y refer to the faces of cell face control volume a, whose normal area vectors at time tn are (A”)Y‘. In computing (uA)aE the A“ that do not coincide with regular cell faces 38 are eliminated in favor of the regular cell face areas as.is.describetinS@-. -aboveve. F-or= faces.y thatlie-emtirely- within.a.regular cdl, the pressures-and valu~-af @p a-retaken-to be tbe~~a~.th~r~~lar.~~~l .u~nte~. F~~fac~- y th~t-lie-on-a-re~lar ceil “face, the pressure. on. face -Y.isobtained-by. averaging.the values.of.@B + (l– ~~.,)p~associated witithe regu.- lar-c~lls cmeither. side of.the cell face. Itis neeessary ‘mdefi~~e--arrad-d-ed-massSanassociated with the spray dropids in cell= face control “volume a. This quantity is given by

, “ .~—a+.$+i+ ~fi-c+ &s;. . s —— (iWf”. “ (My +-(M’);+ (M’)gc + (34’):u

‘ToappmXirna&-d&d.IkEq..(82 we.have used”an -are-admnge-basdon thtkiie kvel”n vertex velocities,

(83) .A:= -Ai (d! -!- t!nh) ; that is, the areas A-~~are-computed usingEq. 158) withlhevertices lo~mtedatx~-+ u~ At, 4. Internal Energy Equation. The Lagrangian phase finite-difference approxi- ma$ioa .~tothe internal energy equation- is%iiefcdlowing:

-t (1 – @Ja (u’’):vu”l..k_v;k + ~ K: V!@_DTB+ (1.– @J fio “A: a

(89)

Tine temperature used-for tlie heat conduction calculation is a weighted average, using- variable implicitness parameter @~, of the Phase B temperature TB and an intermediate temper*”e.T that Wedefineslmr$ly. ‘The Phase- A ifiternai-energy IA”contains changes due.to-chemicallieat.rdease and. the spray energy source:.

Cwi M:kI;.k – M;.kI;k = v!&(Q;k + Q;k) . (90) At

The chemical source Qtikcis given by Eq. (21) with 6, replaced by (ch~A)tih.The method for approximating (d.I~A)U~is described in Appendices I and J, and the difference formula for Q’ is given in Appendix C. The temperature $ used in the diffusion term differencing is based on an internal energy that also includes updates due to enthalpy diffusion and turbulence dissipation:

(91)

To calculate T it is assumed that all heat addition up to this point in the computational cycle has occurred at constant pressure. Thus the temperature change is related to the enthalpy change by

(92)

where

(93)

By substituting for h~kt in (93) using (92), solving for ~~~, and using the fact that h~kn = I@n + pQknlf3gk n, we obtain the following equation for T~k:

1 ?..k = T~.&+ —

By using (89) - (91) we can derive

I:k – I:jk P;k + P;k V:k – V;k M:h + (1 – AO)[@~u (U*): vu B + (1 –@D)du”):vu”l At ‘– 2 At

(95)

40 an equa~icm -that=is-solvwl in Z%aselhmd ‘-willbe-referred-=tc--when we-desmille- the solution procedure. I?cnthe differencing of-the viscous stresses o and the velocity gradients Vu in [~; -thumdti.iir&d:fa.@@iz-K=. “Si Turbulenc@fmation~. Tile-Eagrangi2mpfiase diffhmce approximation %o the inmbuiemtkinet.ic energy equation (7) ‘is

v?.. – wk. M:kk;k. - M;kk;k lJk =– +p;k At ~[jk)k~k + L?8”]-+ (VDI i(~ “– ‘Ljk ijk ijk At

The difference approximation to the dissipation rate equation (8) is

‘The quantity f~h is zero-or unity depending on the sign of the cell volume change Vijp – vgk~:

~- if @ –v;k>o (98) f -= [ ijk ‘ijk. ~ 0 otkerwise .

ThiS prescription for f~h is chosen to avoid”negatitie computed-values of k ands when there are l-kqge-vohnne cha-nges during the Lagrangian phase. Viscous dissipation of mean flow kinetic energy is represented by the term (VD)~h: The difference approximations to the viscous stresses o and velocity gradients VUare given in Appendix K. In differencing the diffusion term, we use a weighted averaged of the Phase A values kA and CAand the Phase B values k~ and cB. The amount of implicitness is specified by the parameter @, which is chosen in a manner described in Appendix H. The Phase A values kA and @ differ from the time n values because they include the effects of spray source term %s:

(loo)

and

(101)

6. Volume Change Equation and Equations of State. The volume change of a computational cell in the Lagrangian phase is computed using the Phase B cell face normal velocities:

(102) V:k = V:~ + At ~ (uA~ . u

This approximates the following equation for volume change of volume V moving with the fluid:

Dv —= u.dA, (103) 19t \ s which can be derived using the Reynolds transport theorem.50 The equations of state (10) and (11) are approximated by

(104)

and

4.2 (106)

‘TheI?i%e A eak!fi%timi i3f&3@3t, plSitiOilSi.ddind-’Uy

(107) where 8X’P is a random position change that is added when timestep At exceeck-the-pm-. title-tmbde~ce..eurelation .time-t~~r~.givemby-Eq. <37) with all .quantities+va-lua%d at timdewl n. The dispkwement8x’P, and -an-associate*ve-lo-city-chamW-8v’D2. are chosen from distributions given in A-ppenciix G. The partidie then exchanges mass, momentum, andemergywith the gas in the computational cell .inwhich .xDAis-located. ‘T15~PHase-A-cai~tiiiiti6n of the ~remaitiiirgparticl~quarltit ie=is-azcompiished by splitting or differencing sequentially the terms associated with each physical effect, always using the most recently obtained droplet properties when calculating changes due ‘~ the next physical effect. The random velocity changes h’p are first added to the particle velocities V403:Then the droplet oscillation and “breakup caifiui~tl~n~i.m~f~~e~(~~- Appendix F), followed by the droplet collision calculation (cf. Ap~_endix E). The Phase A calculation is completed with the updates of particle radii and temperatures due to evaporation (cf. A-pp_endixC) “and”the addition of ~avitationai-acceleration terms to the par. title velocities. The only particle properties that are altered in Phase B are the particle velocities. The Calculation of VPPis intimately- connec-ted.\vith.the.-'dua4ionnf.the.terms..R~~~.and S’”~~,and hence is described in ..Appendix-C-. None of the particle properties are altered in Phase C. n. Li. ScduticmPro-cedure- for YhrmiicitPhase B ‘Equations. TA.Phase-Ilvduesd the fl~~fieid~~~r~ab~es-ar~ fk)~~dbY ~~~’+ing-th~im~]i~it equations of the previous section. ~e.solution.proced~e.i=Lmttanetit~.theSI~LE

43 method, 19with individual equations solved using the conjugate residual method.20 In this section we describe the solution procedure and give the roles of the various subroutines that are involved. Basically, the SIMPLE method is a two-step iterative procedure. After selecting a predicted value of the Phase B pressure p~, in step 1 we freeze the predicted pressure field and solve for other flow quantities using finite difference equations that difference the diffusion terms implicitly. In the original SIMPLE method, the convection terms are also difference implicitly, and their effects are included in step 1. In KIVA, convection is calculated in Phase C in a subcycled explicit fashion that offers some significant advan- tages over implicit methods. In step 2, we freeze the values of the diffusion terms obtained in step 1 and solve for the corrected pressure field using equations that difference pressure terms implicitly. Sometimes a Poisson equation for the pressure is derived and solved in step 2. In ISIVA, for step 2 we simultaneously solve the cell-face velocity equations, the volume change equations, and a linearized form of the equation of state. By algebraically eliminating the volumes and cell-face velocities from these equations in favor of the pressures, one can show we are also solving a Poisson equation for the pressure in step 2. l?ol- lowing step 2, the predicted and corrected pressures are compared. If they agree to within a specified convergence tolerance, the equations have been solved, and we proceed to Phase C. If the difference between the pressure fields exceeds the convergence tolerance, the corrected pressure field becomes the new predicted pressure field, and we return to step 1 and repeat the process. Each pass through the two steps will be called an outer iteration. In Ref. 54 it is argued that one should be able to stop after a small, predetermined number N of outer iterations and have a sufilciently accurate solution of the equations. Such a solution procedure is noniterative and therefore very attractive, but we have found the argument to be flawed. The argument is based on the fact that each outer iteration increases by one order in At the accuracy of the computed approximations to the exact solution of the finite difference equations. By stopping after N outer iterations, one introduces temporal truncation errors whose formal orders are equal to or greater than other truncation errors of the difference equations. We have found through computational experience, however, that it is better to iterate to a prescribed convergence tolerance. Although the errors incurred by stopping after N iterations may formally be of high order, they can be unacceptably large in practice. Because the equations for (Ym) B, kn, and c~ are weakly coupled to the flow field solution, these equations are not included in the outer iteration. The mass fractions (Ym)~ are used in the calculation of the Phase B pressure p~ in Eq. (104), but the values of the Phase B pressures and velocities do not influence the solution of the (Ym)~ through

44 13qi.(T?-).,‘Thus .F&.( .7-7.;& -Sdved=”fklrthe”(“Y~.)‘%fi-subro~titie Y$XXVE%e15Fe begirming tkie.outer iteratiim. ‘I%is results in a ccmskkwabk cm-nputatiomd time savings over schemes, Sm3has those that~=lculatei~icit-convection in step_i; that include the mass f’km.%ion-equa-tionsin the outer iteration. ‘We often have ten or more chemical species in our. applications and to scdv43equations:for M %heseinthe outer ik.ratkm :wd~-g~-atly ifi~rea% “CXIZTIpUhtiOWd~ “~kTf3S h the cases of”k’9”and-cBjthe flow i%ek-influences their values through the turbu- kzme- production-terns-in Eqs; (96 ) and ( 97); but the values of~ and -#%u not-emterinto- the flow field equations. Thus Eqs. (96) and (97) are solved after completion of the outer iteration in subroutine KESCILVE. The finite-difference equations have been designed, of tours% ‘mgivethis-one-way ~mtipling. Mathematically; the-vaiues-of-k and 8 inrluence the flowthrough the turbulent diITusivity and “the turbuient pressure #@k-.Thiii coupling could be accounted for by using Phase B values of k and c to evaluate the turbulent diffus- ivityand jjp~ butthis -would-greatly incm-ase-computai,ional times, is not=necessary for stability, and is usually not necessary for accuracy whentimesteps are.lused.that satisfy the constraints of Sec. 111.H. Thin-the-only equationsin.the outer- iteration.are the momentum equa-tioa; internal energy equation, and-the pressure equation. We now describe in more detail the outer iteration. The predicted pressure ptik~is first initialized by linear extrapolation using_the l?h~se=ll:pr~sures-frorn .%he.previow-two eye-lQs:

This extrapolation has been found to reduce both the number of outer iterations and the number ofiterations required”to soive the pressure equation. The fIrs@quatiomscdYd h-..step -1 .i=.tl=. .J.“ .~e~a.tu-m.ev~a.ti~.... {78). T.% p=die++ pressures ptih~are used in place of the unknown Phase B pressures, and predicted velocity

field u~j~.. is solved “forin piiice ot the Phase B velocities. This calculation is.performefin. subrcmtineVSOLVE. T-}m--~e&cte~temperature fieki is next found “byusing a combination ofEqs. (9-5), (105), (104), and (76). Equation (105) is used to eliminate 1~ in favor of 7% in (95). E.quations

45. which is then used to eliminate V~hB from (95). After some rearrangement and replace- ment of superscripts B with superscripts p to denote we are solving for predicted values, we obtained the following equation for the predicted temperature field:

+ (1 – Ao)[(#)Dduq: Vup+ (1 – @l))O(Un) :Vuniijk II

} (110)

This equation is solved in subroutine TS LVE. After solving for the predicted temperatures, the predicted cell volumes V~hPare found from (109) with superscripts p again replacing the superscripts B:

M! (Y ): wk = +to X* Y’;k. (111) P;k m m I

Finally, we solve for the corrected pressure field p~hc. This involves simultaneous solution of Eq. (86) for the cell face velocities, Eq. (102) for the corrected cell volumes, and a linearized, isentropic equation of state that relates the corrected pressures and volumes. In Eq. (86) the corrected pressures are used on the right-hand side of the equation in place ofp~. In (102) the corrected volume v~hc is used in place of v~h~. The equation of state is obtained by combining (109) and (110), neglecting the diffusion and dissipation terms in (110), and linearizing about the predicted cell volumes and pressures. The result is

wk (112) v;k=v!lk-+— ‘~;k – P;k) , Yljk p~k where

n (Y”,):k ~,_ v.. ~ , lJ k 2(cu);k + 1?” > y 1 I m m V:,k (113) Y;h ‘:k + ‘;k ~. y (Yn,):k - 2(c,):Jk+ _ w P;k Im rn

46 T}ze-com-cte&pmssures-are-found in subroutine I?N3LVE, ‘Tile predicted and-corrected-pressure fields are then compared to see if convergence h-as-been a-tteined. Th~uter-iteration isjmlged’t have cxmverged-if

for. all Cells

~i Phase G Phase. [Cis the re.zxme~hase, ~n-which -’We.Sa-leU12*e$h-nYeeii5e=t~*SpOti=amf3ei- ate~withmcwing.the .mesli relative to the fluid. T-his-is-accomplished in a-subeyeled, ex. plieiteal~dlationu ~ng atimest.ep A& that is an-integral tisubmultiple-ofttre main-comp: utatiuna-l timestep-A-t: “Tinetimestep A-tCmust satisfji the-~nrantr..nditibn .ufAtC/Ax.<.1., where Ur ii+the +Iuid-velimity -reiative-to-the grid velocity, but-’because there is-no upper

!mwadoathe nurnberofsubeyeles, th~~~d~~~I~i~~-titb-~rAt/A~ >1: Tine-exactfbrmula- for -A-tcis-given in-the next section. Convective subcyciifig saves computational time_ ber~use the rezone calculation .takes~ulyabout ten percent of the -time of the Lagrangian p-h-aw~aletilation. Tine user of’IIA-lThas the option ofusing one of-two convection schemes: quasi- second-order upwind (QSOU) differencing, described in Appendix M, and partial donor cell O?BC) differ-mming; described- in-Appendix N. In addition, when using PDC ‘differencing, the user can vary the..=o}mt~ti-’’lapwi ndi>g’> through two P_aramete.rs~~a~Id-fl~.. W-hen a. ==1-and=jjl.-==0; dmmreell or=full up~’inddiffereneing is used: When-aG =0 -anti Po = 1; i~~te~olate-&knor-cell Wferencing, or Leith’s method;%5is used. ‘The algorithms are fully descr~Dedin.t’ne*m&-c+ ~Dut.herewe wHI derncmstrate some offheir- proper-- ties through a computational example. Because ofpoorresolution, the example problem is a severe test of”the convection schemes and exaggerates many of their shortcomings-

47 In the example we convect a scalar field through a two-dimensional mesh of square cells with a uniform velocity directed at a 45 degree angle to the mesh directions. The initial conditions are plotted in Fig. 6. The scalar field is initially unity on a square that has five cells to a side and is zero otherwise. Also shown in Fig. 6 are the results for five different convection schemes after one timestep At such that uAt/Ax = 5, where u is one component of the velocity, The five schemes are subcycled explicit QSOU, donor celI, and interpolated donor cell, each using uAtJAx = 0.2, and fully implicit donor cell and the QUICK scheme,5G both using uAtJAx = uAt/Ax = 5. The latter two schemes are not available in KIVA-11, but are included in the examples in order to illustrate the accuracy of commonly used implicit schemes. Given in Fig. 6 for each convection scheme are contour plots of the final computed scalar field, the maximum and minimum computed

Qc1

Sou Donor Cell uAt. ?Ax = 0.2 uAtc/Ax = 0.2 Max = 0.’87 Min = 0.0 Max = 0.63 Min = 0.0 Error = 0.36 Error = 0.53

Inter~A~~A~ Donor Cell Implicit Donor Cell Implicit QUICK c = 0.2 uAt. \Ax = 5.0 uAtcjAx = 5.0 Max = 1.41 Min = –0.35 Max = 0.’51 Min = 0.0 Max = 0.66 Min = –0.05 Error = 0.41 Error = 0.83 Error = 0.75

Fig. 6. Isopleths from calculations of convection of a square-shaped scalar profile. 48 values,. and the root-mean-square error between the computed swlution and the exact solution; wxhie’his j_@ a -anifomtm~~siati-on-f~ve-ceii-s-in eac’h direction. The.ermmisaverag_ed: over the -twenty-five cells where the scalar field-is-unity in the exact so~ution. ‘Thernost acmrate method-i-s @3t3bT, even-though-it-km T?rst~order sp.atial truncation errors and therefore is formally less.accnrate.than~ nterpdated-dcmor cell: which .is-se~~nd,mder. accurate in space. T=helea-s%.aceurate,methods-a-re theirnplieit methods. This illustrates the fact that although the imp~icit methods are unconditionally. stable, they can have large errors-when-tihk> L. Amonotmm xcimme has the property that it introduces no new mimima or maxima in the computed solution. Tfie donor cell and Q,SOU schemes are monotone, .and.th~their mnqmtdmaximaand-.minima Jie between those of the initia-1amditiomx Interpolated donor cell and QUICK are not monotone. C.omp.utational oscillations.in r~gions ofsteep. Wadierits are obtained with nonmormtone schemes and canhe.especially pronounced i-n calculations using pure interpolated donor cell. Although PDC differencing is less accurate than QSOU, we retain it as an option in fiTA-H-lmcause-ittis-si~ificantly faster than the QSO’U scheme. W-e recommend “useof- P33G diffemlcin~n-calculations in which speed”is more important than accuracy or in calculations in which cell Reynolds numbers are less than two. In the latter case physical di~ffiusionis large enough to render negligible the numerical errors associated with PDC differencing. When using PIIC diflerencing; we recommend taking Do = 1.0 and a. = 0.1 orO;2 ‘to suppress -computation a-ioscillations. Tine QEKYU”scheme shOuid “be used”to oijtaj n the most accurate caicuitititm for a given mesh”resolution. Both schemes can be used in separate caiimlations-of the.samqrnldern+s a.paxtjal .te.st.of.eanv.er.genee. .If.tb.e CQWA-- pute&salutioms-are-thes me, then numerical errors associated “with convection are sma. 1: We now d%cribe the convection calculation in more-detaili The transport of cell- ~mntered quantities-i se]l-fa~. velocities and the old- and new-time ~gridpos;tions:

6vGAt !Al – .&4)BAt 8va = a c. a_ c“

49- In (116), 8VaG is the volume swept out by cell face a when the four vertices defining the face are moved from their old-time positions xijknto their new-time positions Wjkn+ 1. The /5VaG,which are computed using Eqs. (56) and (57), are positive if the volume of cell (i,j,k) is increased by the grid motion. It can be verified using Eq. (102) that the 6Va satisfy

(117)

where NS = At/AtCis the number of convective subcycles. The species densities (pm)~k” after uconvective subcycles are given by

(Pm);y’.k = (Pm);;y’.; 1+ ~ (pm):- lWO, (118) a where the summation is over the faces of cell (z,j,k). The species densities are initialized at the beginning of Phase C by their Phase B values

(119) (Pm);k= (Pm):k , and the subcycle volumes Vtik” are given by

V;~ = [u V;; l + (NS – u) Vf!k]/Ns , (120)

where NS is the number of convective subcycles and Vtikn+ * is the cell volume based on the final coordinates. The cell face densities (Pm)auare evaluated either by the quasi- second-order upwind scheme described in Appendix M or by the partial donor cell procedure described in Appendix N. The total density after u convective subcycles is given by

P;, = ~ (Pm);, ) (121) m

and the vertex masses after u subcycles are found from Eqs. (66) and (67) using these total densities of (121) and the volumes of Eq. (120). The specific internal energy ~~k”after uconvective subcycles is determined from

“-l U–I.–1+ ~ (pna (122) ‘~k ‘~k~jk = ‘ljk ‘Qk lijk ~ ‘-’tiv u’

50 wl-we..therd face ermrgy.dimsities

The f’armula for updating turbulence quantity qtik” in the subcyele-is

,1 *.-. p;kv;.kq;k =p;; ‘Vljk:–1qijku-l”+ ~ (pq):-iwa, a where q = k or q = k3’2/c = L. We convect turbulence length scale L rather than c be- eaus+ee.gem%%dly=has.s!!eep.: ‘gra&”ent&-.:m&ihe4wfm-la3~:mmwi-a31:eYrors-a-r:se -w-ken‘ comvm~. W:cellke.quan.ti.ties (pq)au are evaluate d cit.her by quati%eccmd-ordsr differencing (Appendix M) or by partial donor cell differencinglAppendix N) using-the. full. --.--—.. . donor celi limlt (aO = 1, ~~ = O). Since partial donor cell differencing is not monotone exwpt=wheruo = land Do = 0; this limit is Used-Atensure ‘tkat-rlegative-va-lues ofk and”L ammot:cdbtaineci in the convection phase. After completion of all convective subcycles, the final values of cell-centered quantities are set equal ‘m their-values aftert&NS- subcycles: Tine final value of temperature is computed by inverting J@ [113 lusingfinal “values of%ternal energy an&rrass&nsit.ies.. ‘I’lie final pressure is given by

~+i-_ (124) P~j~ – ROT;,:’ ~“ (pm);;;/ Wm . m

Convective transport of momentum on subcycle. U.is calculated in terms of the mass increments across momentum cell .fac.esIwhit-h an+related to the mass increments across regulm-ce-ii faces in the following way. ‘The mass increment across ceil face a of-a particular momentum cell-is definedliy

where o and i are the regular cell faces on either side.of .the.momentum.cell -fa~. a. (between which face a is ‘

(127)

where the velocities up”– 1 are evaluated by the quasi-second-order upwind scheme of Appendix M or the partial donor cell scheme of Appendix N. Two special features are provided in conjunction with the convection calculation. The use of partial donor cell differencing sometimes results in the development of unphysical small negative species densities. We therefore provide a reapportionment algorithm which tends to preserve the positivity of these densities, as described in Appendix O. This algorithm is not needed when quasi-second-order differencing is used, since this scheme is monotone. In an axisymmetric swirling flow with free-slip boundary conditions, the total angular momentum should be conserved. However, the K.IVA difference approximations to the momentum equations simply conserve the three Cartesian components of momentum, and this does not imply angular momentum conservation because of truncation errors. In practice, we have found that the only such truncation errors that are significant are those arising from the rezone calculation of Phase C. For this reason, an optional angular momentum correction procedure has been included in Phase C, as described in Appendix P.

H. Accuracy Conditions and Automatic Timestep Control The timesteps At and Atc are selected automatically at the beginning of each cycle. Because diffusion terms are difference implicitly and convective terms are subcycled, there are no stability restrictions on At, but there are several accuracy conditions upon which the automatic selection of At is based. These will be given in this section. The convection timestep Atc must satisfy the Courant condition for stability, and we also describe how this is generalized to an arbitrary mesh. The accuracy conditions we use to determine At cannot give a universally reliable selection because there are many accuracy conditions we have not taken into account. It has been our experience, however, that the criteria we use for determining At give tempo-

52 rally= aeelura-tesdu tiorwi n most eahmlati mm T+heuse:rismamtiimm‘ -f&t’ Oihe~ a-ccu.rac.y conditions could be important in his application and the timestep shoddalways.be.varied ta .test.ftir.temporal accuracy= The.&r~t a~~~rae~ ~~~ditiofi on At is that,

(128)

where ,Fais swne positive real number oforder- unity and Ax-is anaver=ge--ceil dimension. This.ccmdition. arisedecau-adermsdo rder higher. than At.are igncweAh-Eq-(115),. The accnracy. cmxstraint< 128) is the only-one we use in which the cdl sizeAx appears. We note

that Ai - Az@w eendi tie~.-(1-28)j in -Cofitrftstto-eqli d C%%lVeC..k’e--SM3~lity.-cz=ittnri-zqW’hic’n give At- Ax, and exp~icit diffusional stability criteria, which give At - AX2. Thus while (128) will reduce the timestep as the mesh is refined, it will not do so as much as the two stability criteria, which bad to-be observed by theflrst version ofKfVA.~ Constraint (i%).is implemented by.calculating a timestep Atacc:

and the subscripts-in (lillqj refex=ta.the=ver~i~~sof ~mll(i,j,k) as numbered in-Pig. 2: We- then constrain At.fi+ ] h kwlkss.ttan- Atticen-+1; as described below. ‘Ike default va!tie-of !ra_ is 0.5. The second accuracy condition on At is that

IAI At

where fr is oforcler unity-and A is an eigenva-lueofthe-rate ofstraintensor. This criteti- on-lhnits theanmunt-of-deil ai~tortion that can occur due to mesh movement in the Lagrangian phase. -When.ceJls hemme very-distorted, the spatial accuracy of thediffer- ~nc.e ~p~_~ox~m~tio~s.~~ff&~j one ~xfHI-Iple ofhow (13-1) ‘w-or*SisthKf~ilOting.” In .apkme parallel shear flow, there isone nonzero eigenvalue.+ tlui~y,.wherml~is.the sdreamwise ve- Iimity component amdy is the crwss-stream direction. As-depicted” in Fig. 7-for a rectangu-

— 1------I I I Fig.7. Cellshapes before (dashed tiy : lines) and after (solid lines) I Y 1 i t the Lagrangian phase in a 1 I plane, parallel shear flow. 1 / I /

lar cell aligned with the flow, constraint (131) limits the distance upper and lower cell vertices can move relative to one another, divided by the cell height. Constraint (131) is enforced by calculating a timestep AtrS~:

f, (132) At~sl= min (i,j,k) 2d~ ‘

where

(133) aijk = ~(p~k – 3qtjk)

and pijk and qijk are given in terms of the rate of strain tensor sem in cell (i,j,k):

(134) PIJk = ‘see

+ et.ms1es3nL + cem3s1es2n, . ‘LJk = ‘l f?msZt%n

In (134) C{mnis the alternating tensor,57 and stm is given by

~ due au —+J (135) ‘t’rn= ; ( ax~ dxe ) ijk ‘

where the velocity derivatives are evaluated using time level n velocities and the approximations of Appendix K. It can be verified from the formula for the roots of a cubic polyno- mial that the denominator of (132) is greater than the magnitudes of all eigenvalues of Stm, and therefore if we select Atn < AtrStnthe constraint (131) will be satisfied. Using f.= U=has given sufficient accuracy in calculations we have made. Two other accuracy criteria for At are obtained from the need to couple accurately the flow field and source terms due to chemical heat release and mass and energy exchange with the spray. For the chemical heat release, the requirement is that

54 Vtik (i.. “iJ”k. Llt <-fch , MGkItik where fc~ is an input constant typically taken to be 0.1. Constraint (136) is the requirement that the total heat release from all chemical reactions in a cell should not exceed a small fracticnr~c~ ofthe-totzd internal energy in the cell. “Toenforce (136) “we caicuitite timestep A-tc}tby

and choose A-tn-+~”~ A-tC~E-+i. ‘This formula assumes that the fractional rate of energy reiease varies SIOwly from cycle to cycle. Spray timestep At~Pis calculated. from a similar formula:

‘~-~R-”I~diU-~~m~Sk~” used for cycle n +1-is then given by

Atn+l = ,mk(Ab~~l, &~~~l, At~h+l ,At~P+l, Atn+l, At;X, At ) . s I?LXcc

T=h&imestep Atgrn+ 1 limits the anmunt=by ‘which-the-timestep can grow:

At rt+l _ (139) @ —1 .Q2 Atn .

‘Tirnesteps Atmz and Atmxcaare, respectively, an input maximum timestep and a maximum timestep based on an input maximum crank angle in engine calculations. The-initial guess for Aioncycle O-isgivemby input-quantity llTL Tnis is then compared with Atrsto, Atmx, and Atmx-a to determine the actual initial timestep At”. (If the IYITsspplied--on a-subsequent restart ditffers firomthe DITat cycle O; the current At will.be. reset to this new EiTE) T-heconveetion timest-ep Atc k based-on-the--Courant-stability condition. in a rectan- gmlarmesk, this condition is

~z . ) (140) ‘ [w–hzl~’ where & bY,and bz are the components of the grid velocity b. Constraint (140) limits the magnitude of the flux volume in any coordinate direction to a value less than the cell volume. To generalize this to an arbitrary mesh, it is natural to replace (140) by the similar criterion

(141) where the i5Vaare the flux volumes calculated for cell (i, j,k) using timestep Atcn– 1. In practice, for accuracy we also reduce the timestep determined from (141) by a factor fc~u, typically taken to be 0.2.

IV. THE COMPUTER PROGRAM

A. General Structure The KIVA-11 computer program consists of a set of subroutines controlled by a short main program. The general structure is illustrated in Fig. 8, showing a top-to-boti%m flow encompassing the entire calculation. Beside each box in the flow diagram appears the name(s) of the primary subroutines(s) responsible for the associated task. In addition to the primary subroutines, Fig. 8 also identifies a number of supporting subroutines that perform tasks for the primaries. Comments at the beginning of each subroutine in the listing describe its purpose, where it is called from, and what subroutines or functions it calls, if any. KIVA-11 is an advanced experimental computer program, not a “black box” production code. Its use requires some knowledge of and experience with numerical fluid dynamics, chemistry, and spray modeling. ICIYA-11 was written specifically for use on the CRI Cray family of computers, oper- ating under the Cray Time Sharing System (CTSS) and using the Cray FORTRAN (CFT) and CFT’77 compilers. We have several observations concerning our experience with the CFT compilers currently available to us: ● CFT 1.11 k the compiler of choice for short runs and scoping studies using KIVA-11. It will compile the program in about 20s on a Cray X-MY, and the compiled code will run the baseline sample calculation in about 63s. ● CFI’ 1.14 requires about 55s to compile KIVA-11. The baseline calculation still requires 63s to run. A principal feature of CFT 1.14 is vectorization of loops containing indirect addressing, utilizing hardware features of the Cray X-MP. While this is quite attractive for some of our codes, it is of minor consequence in KIVA-11, which contains very little indirect addressing. 56 * CFT77, ‘written-inPascal rather than machine kmguage, requires over 3-rein to compile-~A=btish~t.r~er.ved .f~r.long&rruns In which one sa-nrecover tbcorn-- pilationtiinepenalty. The baseline runtime is dowrrLtabout-52 ‘s; 21% faster than with*he-CFTcorr@hws. Clearly, it is advisable-in create.tiibrar.y of~docatabi-e bi~ nari~.when using CJCTL D4.andXMT7.7j +Aminimiz~th~tim~~pmt-in t,he~mmpiler. At present, CITI’77 is not a good compiler to use for debug@ ng,.as many variables reside in registers. Therefore, symbolic namesare.freqmmtly- inaccm.ssible to the debuggw. W.e.m&rOperating System .(COS) and.willfindK~J.4-Hg~ ~aJly compatible. Tile.prirmipk incompatibility with outside Cray systems lies in the calls that communicate with the opera-ting-syskxn. T-he-functions of all ~mdls‘tosystem-muti~~s-are-d-esctibed-in the EI?HXX3 at-the end ofthe FGRTRAN listing. ‘WRITE “(59;-) statements refer to the user’s remote terminal. Users who do not have a C!ray computer face an additional task adapting_K.I_VA-H to run on whatever computer they have. This is because K.IVA-11 contains statements pecul- iar to the (WI’ compilers that permit vectorization of many of [email protected] sub. ~routines. Oul~ffotis-at-vectotimtion resulted in making the hydro portion oi’the code run nearly five times as fast. (Unfortunately, .the.chemistr.y. and-spray-subroutines are not amenable to Sue-htreatment; this is because each cell in the chemistry and each droplet in the spray follows a unique logic path dependent upon local conditions.) ‘The unfamiliar statements in KD7A-IFare the CFT vector merge functions CVMGT, CVMGP, CVMGM, and CVMGZ, and the vectorization directive CDIR$ IVDEP. The vectormerge-furmtions-allow many loops to vectorize, in that they can repiace IF ‘tests, ...... wlmh cmnot vectkrize. 13riefTy,tlie-fourveetor merge functions perform as follows: > CVMC-’T (X-,Y-,L) results-in .X-if L is true, Y-ifL is nottrue. * CV.MGP {>~,xX’,P)~es~~tsin X if”p’>— ~, y ifp <9. @ CVMGM-(X;Y,M) results in X-ifM < O“,Y-ifM —>0. s G’VMGZ {X-,Y-,Z)-results in X-if Z-is zero; Y ifZ is ilO1~TO; CX31R$IVDEP instructs-Cl?T to ignore apparent vector depend&ci@s mrecursiims. Ifk.mhnagineckxmrsion.ca uses the 10-opndto-beautomatieally vector%e&by CWT, the

~v~DE~dir~~i~e -instructs the ~mp~l~r~~ ~reate ve-cturto-&-aHyway: NO~~ that thiS directive begins in column 1, which will cause it simply to be treated as a.comment by other compilers. START PRIMARY SUPPORTING 9 SUBROUTINES: SUBROUTWES: BEGIN

\ READ INPUT. COMPUTE DERfVEDQUANTITIES RltWUT, TAPERD

CREATE MESH AND CELL VAMABLESI SETUP, TAPERD VOLU15 PISTON, k STATL BC

~ CALCULATE GAS WSCOSITY I Vlsc I CALCULATE AREA PROJECTIONSI APROJ I I CALCULATE At FOR NEXT CYCLE I TIMSTP

] PEW CYCLE: WTPUT, CELL lNITIALiZA~ NEwYc DMPOUT, FULOUT, p&t, print routlms I TAPEWR Imo 1 MOVE P{STON, CALC. ITS tWW VELOCITY I I 1

~ NJQ2T FUEL DROPLETS, IF APWOPRIATE INJECT FRAN

DROPLET TRANSPORT AND TURBULENCE\ PMOVTV FRAN, WIND, REPACK

I DROPLET BREAKUP I BREAK FRAN

I DROPLET COLLISIONS / COALESCENCE \ COL~E FRAN REPACK

[ DROPLET EWPORATION ] mP

WLL SHEAR STRESSES, HEN TRANSFER LAWALL BC I ! OPTiONAL NODE COUPLER NODCPL BC

KNEllC CHEMISTRY,tGMTION CHEM ,,I EQULBRIUMCHEMISTRY CHEtmof (tnocr systemsdvor’) cwQGM I, [ BODY ACCELERATIONS GRAVITY BC

PwM, BC PCOUPL

Fig. 8. General flow diagram for the KIVA-11 program. 58 PRIMARY SUBROUTF4ES:

ml?wic

[ MASSDIFFUSB2N I Ysfx!a YIT,.RESY 1 Exfm= Bc-

Pwr i ADD PRESSUREACCELERATIONTO VELOCITIES i PGRAD BC, APROJ

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D=QQ~. ,. Be=.APRoJ I i

[ ADO PRESSUREACCELmATION TO VELOCtTIES I PGRAD BC, APROJ

PHASEW j :ikfPUC4T-DIFFUSIONOF”TKE-APWEPS ] KMW DRllKc REsK. REsG BcEm BcRE$E j UPDATEDROPLET VELOCITIES !.= PACCEL . . 1

~ CALC’UEATE-me vE+,.oems* j RE20W I REZOKGMD COORDUWES I I voLLJii4E

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, @mmxETsmNEABovE- [ CNOP BcrBcEPsr DoML I tF A?PRomATE ?FM, vokuME

STATE

Fig.- 8. continued 59 The non-Cray user can quickly modify the program by installing functions that emu- late the vector merges. For example, CVMGT can be emulated by FUNCTION CVMGT (X,Y,L) LOGICAL L IF (L) THEN CVMGT = X RETURN ELSE CVMGT = Y RETURN ENDIF END Once the program is running properly, the functions should be replaced by in-line coding for greater efficiency. Overall subroutine architecture also shows the effects of vectorization. For example, the user may ask why we break some logical task into a set of contiguous separate DO loops, when a single DO loop would appear to sui%ce. The reason is that excessively long DO loops will not vectorize because of optimization-block size limitations in some CFT compilers predating CFT77. It is not easy to define just how long ‘(too long” is, as it depends not on the number of statements, but rather on the extent of computations involved. These cases were determined empirically; if there was no other reason why a long loop failed to vectmize, we broke it up and achieved vectorization. The dimensions in the release version allow up to 12 species, 966 vertices, and 2000 computational spray particles. These dimensions may easily be altered via the PARAMETER statement (lines COMD.18 - COMD.19). The input quantities always required to setup a problem are described in the EPILOG at the end of the listing and are read according to the formats appearing in subroutine RINPUT. The mesh generation is automated for a broad range of engine geometries, and is discussed in Sec. IV.E. I B. The Computing Mesh The KIVA-11 formulation is based on (x,y,z) Cartesian geometry and is applicable to cylindrical (CYL = 1.0) or planar (CYL = 0.0) calculations in either two or three space dimensions. The mesh is composed of a block of cells in logic space, NX cells in the i-direction, by NY cells in the j-direction, by NZ cells in the k-direction. 1. The Five Mesh Types Available. Figure 9 shows the five mesh types available, determined by the specification of NY, CYL, JSECTR, and THSECT in the input data.

60 .. E

~y *

IsEC-m =1

JSECTR I“.—=0

...... - Fig. 9. Tne_five-meA.ty~vailableln.-~=.A-D~ The 2-D cylindrical option (NY =1, CYL = 1.0, JSECTR = 1) offers an effkient means of calculating fully symmetric cylindrical geometries. In three dimensions, the cylindrical case results in a pseudo-polar grid. If NY >1 and JSECTR = 1, the resulting mesh is some sector of a full circle. Figure 10 illustrates a 72° sector, which has been applied to an engine geometry in which the on-axis fuel injector has 5 evenly-spaced nozzles, directed radially. The sector option allows us to model the 5-nozzle feature easily and far more efficiently than zoning a full 360° by taking advantage of the symmetry. The JSECTR = Ooption is used when features of th(’ geometry or spray require zoning the full 360°. As shown in Fig. 11, the Cartesian block of cells is curved around and joined to itself. In all three cylindrical cases, the front and back (derriere) boundaries are periodic. The neighboring-cell relationships between cells facing the front and derriere boundaries are built into KIVA-11, and velocities are mass-averaged across corresponding points. The left boundary is shrunk to zero size to become the central axis, where our prescription at each axial level is to separately mass average each velocity component. For engine applications, the top boundary becomes the cylinder head, which may be flat or domed, and the bottom boundary is the moving piston face, which maybe flat or contain a bowl for DISC or diesel designs.

Fig. 10. Perspective view of the outline Fig. 11. The KIVA-11 3-D pseudo-polar of a KIVA-11 sector mesh. grid is formed from a Cartesian NX = 20, NY = 5, NZ = 10, block of cells through the use of and THSECT = 72°. periodic boundary conditions.

62 The a-zhnuthal dimension oftheeylindrhxd options is-given by T-HSECT, ~measured- imdeg-mes; ‘1’HSECT is-required to be 3tW-for the full-circle mesh, or an even fraction of” 360C for a sector mesh, so that the symmetry condition is satisfied. For example, the mesh in-Fig. 10-has T-HSECW = 72°, as there are 5 -spray nozzles Imingmodele-ck lWmm-NY =- 1-, subroutine ‘RINITT ensures thatTHSECT = 0.50=. For whichever of the five mesh confi~rations-thatthe user ~deetsj KIIJA-H-auto- matieally ~mmptitesthe ccorrec~bmmdmy cdtiom-tmatients-forno~slip, free-slip, or free-slip law-of-the-wall, .requiringno code.modificationdy. the.usw, [email protected]’onspecial. omsessuch as-inflmv=and outflow=treatments; whic=h+mAise=ussed in See. IV. K-. 2. 2-D to 3-D Conversion. Further efficiency in the use of 3-D sector and full circle meshes is made possible through the use ofa 2-D to 3-D converter supplied in KIVA– II. IzTImdrlyan@~aTFlieaiionsj a=si~ifieantportiono fthe cahmiationis the cold’fltnv afterlIVC amd-beforethe spray event, while the flow is truly axisymmetric. ‘Tiiis portion cmheno&ldlting. the L-D.cylindriml option .(NY- = 1)j then .forc-ing-arestartdump just befort%hvsray. begins. The-user- then~~~ti~tirm:t.hisdump; sugplyir~~ane.w-vatie=&f NY and-an a~propriate-- value of THSEC:T in the-input .&t&. WbrnntindUNIZUZnotes. that NY has changed and calls subroutine TRAN3D to convert the 2-D mesh with its current=solution ‘to a 3=13sectururfull circie-mes%--with-the-same currentsoiution. “Tile-3~i3 run then proceeds justas if it had been 3-D ail”along, and the user can initiate a nonaxi- symmetric- spray eventj havingrea-lized a-signifieantreduetion in ~mmpater time to reach” this pointifi-the ~dcrdatiom T-}m--oTm:msttiction:un-thisfedWe is thatilre-parameter NV, which is the dimension of a cell stora~.array, rntik.adeqa~e.frn m.t.imet. = o to aw.om. rnc&te-the.%-Ihnesh. c. The Indexing Notation As-discussed in Sec. 13X13;some.variatiles are.iocded at.vertices and”some at ceil” centers. iii FOR TRAN-notation x~k becomes X(I,J,K), p~k becomes W,J,W, and so on. ‘Thus; th&ndices(&JJQ refer- ~tmthe cell ~~nterforee]l-c~ntemd-va fiaMes-orto-vetiex- (i; j;k) for vertex quantities. I&cause the number. of.vertices inanydiree-tion is one greater than th~numberof ~~.ls~itis=apparent that the-grid in mmputemt-orage n-mst-’b@tX- +-1-)by{NY+- 1-)by (NZ + 1) in-size: Since our index (1, J;K) refers-to both cell-center=~vext.ices, -we -must. allow- extra .storage planes ac-ross -the right j back, and top Qfthdogiealmesh. In K-TVA-II we replace the tri@e_(I,J.K) [email protected]@esubw~ipt, which allows. statements to be compactly written. Traditionally single subscripts have. also be.enmore. ei%ien~, but this isbemrning less-of arvadvantagewitk ti~~inc~~+n-g-’mT-hi~ica%ion of the newer compilers. When referencingthe eight vertices of a cell, .we use the 1 through 8 shorthafidnetiefio fFig. 2. In”this rmtzatiori:‘T-4”r&erstu.ve-rteX4; $hef3;J-;lK3vefiex. l~is

63“ computed as (K – l)* NXPNYP + (J – l)*NXP +1, where NXPNYP = (NX + l)*(NY + 1) = the number of vertices in a plane and NXP = (NX + 1) = the number of radial vertices. When referencing the six neighboring cells to obtain cell-centered variables, we use subscripts with the letter P for + and M for —, when necessary. Thus, we write IMJK for (i – 1, j,k), 11 for (i +1, j,k), IJMK for (i, j–1, k), 13 for (i, j + l,k), IJKM for (i, j, k–l), and 18 for (i, j,k+l). In vector loops that update the 8 vertices of a cell, such as for pressure accelerations or the calculation of vertex masses, note that the sequence is always 6-7-5-8-2-3-1-4. This is dictated by the rule that senior array elements must appear before junior array elements in order to avoid vector dependencies of the results-not-ready or value-destroyed types.

D. Stora~e of Cell Data For many applications, KIVA-11 can make heavy demands on computer storage. Even with minimal cell resolution, three-dimensional calculations require several thousand cells, and the multiple species and spray model capabilities add to the demand. Because we operate today in a time-sharing environment, reasonably efficient use of computer storage becomes imperative. Accordingly, we have equivalence as many storage arrays as possible in KIVA-11. The idea is to retain quantities during a calculational cycle only as long as they are needed, and then to reassign the available storage to other quantities. In this version of KIVA-11, the full calculational cycle requires 217 variables of subscript (i,j,k), plus species densities (12 arrays) and species masses (another 12 arrays). As part of the equivalencing, the species densities and species masses share the same storage. The final storage scheme requires 100 arrays rather than 241, a 59% reduction. Figure 12 shows the allocation of these 100 arrays. The ordering from left to right corresponds to the sequence in which subroutines are called during a cycle. Reading down a particular column, the appearance of a variable name signifies reference to it in the associated subroutine.

E. Mesh Generation Although the features in KIVA-11 provide a general capability, applications to internal combustion engine modeling were the principal reason for writing the program. With this in mind, we have included an automated mesh generator in subroutine SETUP that will create a usable 2-D or 3-D cylindrical grid for a wide variety of piston and head shapes for both DISC and diesel engines. The generator requires the use of tabular information as part of the input data. 64 SUB- I \ RoUTINE 1 SSTWP VISC APROJ TIMSTP NEWCYC INJECT PN@mv mZAK coLIm ZVA?. WALL WM3CPL I voLm!s PVLOUT P?’IND RZPACX. Bc– 2X2 \ ~Em. i I TAmwT- F@PACX \l DuPOUT .AMAY - ‘, : ...... e- .....-m... - ......

x x ------x x x ------.x-.. ------— ~— x- Y Y ------Y Y Y ------f ------. Y Y z 74 ------z z z z ~_z -...:-. -.----;----::::::::::: “ z. .. ---.------z------ti- . ~.“c u-- U Lr u !3 =------: v ------— v ./ — ------v ~. ------~. w w ------g. _~, ?+ =------w w Ro Ro no. ------r!o =------xo– --.---..- Ro RO ------VOL ------VOL VOL ------VOL VQL Vo% VOL. ------.----- .------— p p. ------AK ANv ------Arm ------? T ------? ? ------? ----...... F. ~ p. n’ ------~_Pv ::::::::: --=-.-::::::::::::::::::...;=..-.. 4P ~- ?ZNP &- m- ...... ------SIE ------...... ------SIE SIE ------2X5 ------Txz -..---.-- Tm ...... ------XT%B ------STAS ‘------ITAB ------JTAB ‘------JTAB-~&- ‘------.~~- ‘------— ------.JTAB- .XT3iE ------...... =r’– ------...... ------...... - ...... ------...... ------.s-- ~- ...... -.--..-.--.— ------w UN w ------W.. w- .w- ...... ------.-— ------Bznw SUvw ------Sk s% SPD ...... - Sm...... -..------s-=------...-.--– sFD------

1KVmrw ...... Pnv ...... - Rxv Pnv GAwxA------GAmLA .-...--... -.-.-.----— .--- ...... GAtQ6A GAmIA ------izPs 23PS ------laps ------ms ------ALX ALK ...... ------AM ALY i?x ...... ------...... APx APx AsY,- ~r. ~------...... APz A?Z ABX ------..------&*— ~~ , ABY”- ABY MY ABZ ------As?. .AB3.

E16 E17- Els”

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&J slJa- \ RoUTINE ! CNEN C56ENEQ PUOM Pco(iPL YSOLVE .ZXOIF PINIT PGRAD VSOLVZ TSOLVE \ I (OR Bc YIT Bc (1.0) REsbw TINVRT \ CNHCCM ) RESY sic Elc REST \; APRoJ OROT APJUY \ I ...... I x ------x x x x x x ------Y I ------Y Y Y Y Y Y ------z I ------z Z z z ------u I ------U ------: ------.- U : ------v I------v ------v ------v v ------I------w ------w ------w 54 ------R!I IRo RO RO RO RO RO RO RO RO VOL ...... ------VOL VOL VOL ---~:------VOL VOL P 1...... - P ------P .— ------Anu I Ado AL ----:------:---- Anw Awu 1? ? ------? r ? ? ? r F : 1 ------Fw ------...... ------TZHP IT=?TZ?I?ITZZ’SP ------TENP T~P ------: ------TEt5P S1S! SIX -.------SIE SIE SIE ------SIE I— ------TzCS ------TRS ------mot ------1% — ------ITAB ITAB ‘------ITAB ------ITAB JTAS 1 — ------JTAB JTAB ------JTAB ------JTAB KTAB I —.-— ------KTAB FiTAB ------3(TAB ------KTAB PIT ------PIT “ ------PIT1 I — ------...... PIT1 ------UN I-—-...... -—- UN ------UN ...... -. .-.---- ...... ------w I--—..--.....---- VN ------VN ...... ------NW --.-...---...... NN ------WN ...... ------...... ------Suvw I------Suvw Suvw ------SUV64 ------Suvw ------SPD I ●m SPD ------SPO SPD ------??!!--- SPD El I------Nv,Pnv ------R2tv .-...---- Rxv 3U4V ------E2 1------CA2uA ------GAMA GAsuA ..-.-..------~- ,~~- .....-— ------rue ...... - I ZPS --- . . . :: I ----— ------ALX ALx ------A x A x ------Es -—.— ------ALY ALY ------,Y ------i ..- A ,Y A l!.s -- —— ------AU ...... A .2 A ,2 ------m ,.----—-. ------AFx AFx ...... A x A ‘x ------n ,----— ------AFY .------A ? A Y ------n .---— ------.------AFZ AFz .------A ~z A ‘z ------Elo . ..--- —- ...... ------ABX ASS ------A )% A lx ------Eil .- — ------ABY ASY ------1 iY A IY ------I!12 .------ADZ ADZ ...... - a M A 12 ------EC! CLI Cz.z CLI E14 CLT CIJ ...... - ...... ------CLZ Zls CLK CLX ...... ------CLX EILZ CFI CFI CFI X17 CFJ CFJ ------CFJ Eln CFK C?K ...... ------.- CFK Z19 CBI cBI CBI E20 CB2 CBJ ...... --. -.-...... C5U E21 CBK CBX ------CBK E22 FSUN14 ,RFSUM14 RFSU9S14 RFSUN14 E23 FSUN34,RFSUN34 RFSU3634 ------RFSUN34 E24 FSUM4 ,RFsUNS4 RFSUXB4 ------RFSUN$4 X25 YSPW WDX OUDX E26 YSPD W9Y WDY Cv E27 C4JDZ DUDZ R Z28 XCEN DVDX OVDX SIETIL X29 YCEN DVDY OVDY IE30 ZCS34 DVDZ DVDZ E31 C44DX D41DX CVTERN E32 c4mY 04?DY E33 C44DZ DWDZ PHID PHID ...... ------PNID PHID E34 ------E35 SNTHDF DISPTIL ------DISPTIL E36 SPD14, DYP14 TU14 Msu TZN14 E37 sPD34, DYP34 TSK34 Rssv TZM34 .s38 RU RU SPD84 ,DYP84 TE16S4 2WSW TZWS 4 RV RV HISP UTIL ------E39 ------E40 RW RW SPNTIL VTIL ENTHTIL NTIL ------241 ------E42 00Y Rnvsu ------Rnvsu RNvsu E43 Rzsuo Rzs IZ44 3$%:LO Rs9vo ~sOLD E45 DAM Rzswo ORSS 5246 RDRDY DRSSU RDRDT X47 D!2LTAY DRZSV oTE14P TNZ14 DRZSW E48 ------E49 TKZ34 PN PN PN TKZ84 PNIP P!61P ------Eso ---.---..--- E51 EPS14 UB ZPS34 VB ------E52 ------lx)] EPS84 WE E54 DISSXP DISSIP Hs ------DuTOT DNToT DUHAT HTC ES6 DVHAT ,VOLB VOLB ------DSIEP DSIEP WHAT E57 ------E5a OTNZP oTnP CPC CPC .------.------. ------RON,~~IL ------HTGI’IL E59 ------E60 ------. ---. ---.------SIEN,~IL I TTIL E61 TKZTIL ------E62 EPSTIL ‘------E63 ------E64 ------

Fig. 12. continued 66 --- ROUTINE i PSOLVZ PCMD PORAD PNASEB. KZSOLY PACCEL !?EzowE– V2Z2XS CCHXA xe%2FLx asp 5TxT-z- \ ~i.t: ~fi; ~. ‘.. ! X’22A4.:2.3}. (s:?! SCZ* 2C vem- \ 1BC,APRCLJ,BCFC UC “SC RZSE ,UCEPS BC.BCE2S. .. !CFxifm. FEXF3:7= J@R~– APRG!J L!=REsE= GIQ3u& ARPAY\l MDP ,Rzs P WHZ ,P?IND

x i x x x x x ------x x x x x ------y y. :– “_ Y Y Y Y Y. Y_ ------Y . ------. z z z z z z ------z ------, u u u u u ------u : -~--:------: ---- : u ------v v v v v ------v v ------v v ----..--- *. ,. .“. ~. % “ w ------w ------w ------RO ------RO .mL= R~ RO RO VOL I-----:-:: ----:::::::::::::::~: ---~-::-- Vcz VOL ------P P P P P --??-- --:::::::::::::::::: ”-..::::-- P .------F P AmlJ ------F. ~- ~- Anu Am- F r- ...... r. ~. p. -. ~- ~ r . ~ FY ------...... ------. ------; Pv ------, ------TEMP TSMP_ TEKP S1.2 ! ------BY*- ...... ------SIX ------S1’S S1)3 1 TKE ------‘Tsz ...... ------T= ------I;~B I:~B ITAl! 1% ------ITAB ------XTAS ...... - ITAB ------JTAB I JTAB JTAB JTAB. ------.~AB .--. - ...... =. ---.--— “y~— ------JTXB– ------I(TAB XTAB 3(TAP. Q!&s ------.&TAs - .-. --.-a .-a -—------PIT 1------...... - .-. --. -.. -.-..— ------~“ ------Prm i------P!X ---...--.. .-.-.-... --....-..e ------hTI ---.----- UN IUN ------UN VN I’m ------VN w. Nn ------wl- Suvn Sbi Suvw Suvw -----.-- Euvu SUVN 1 1 ~pD. SPD ------SW ------=------– ------.SPD; SPU SPD

XI P3w Sxv RIw ------..------_-- P31v nv, m ------n ROA)MA ‘------RGNDLA ------mamm ~-– Z3 ------EPS ------—--- me, SCL ------me --.-.---- E4 ------...... e.---..- 7$$. X5, Ala Au ALx AM ...... —.-.. Axx AxN I&Y ~- E6 ?&2 us. AL2 ...... - H-....- E7 APx ------.------E8 A?Y m. ------.-s.---- ASK m A?z AFz . ------.-.----- An E1O ABX Asx % ...... ------.- Asx Ell AEY ABY Am ...... ------Am =J.2- ~z- ...... ------~13. As% ASS– m...... ------CLX El& .- . . ..-... ---= ------G ------Cxs EIS ------CIX E16 ------CFI- ~~7 ------...... ----- c~J: EII ------C?x E19 ------cm E2G ------Cn.r E21 ------CBK ------E22 ------R?SU)114 Xo Xo E23. ------— RFseM34- Y*. ------– W ------E24 ------R?SUM4 20 Zo E25 UAL ------UAL E26- Uh? ------.-.-.---—--- m3L E27 Ins ...... ------..-.---.------.-.---.- E2S RPA ------RPA Xgmn ~23. ~- Ycm ------1 . ------E30 3UI,3WL0’T ZCLS 1 E31- Hs’P3@m- XL ------XL ------= .s32. !!!.PMDT YL =------— Yr4 ------=– = E33 ZL ...... ------ZL ------E34 ------PHXD. FxL ...... PXL- .------Fxi ------E35 ------I ------E36 UAU iii Vou ------you” ------I %% E37 gxFx- VU33” ~--- ,- 23a: ~*- s083E- Vnon VKO16 E39 ------UTXL 3!oSCL. !!no6– H160H– E&o_ =------VTXL ROS13EV SXO16 T’rsx E41 ------Wrxr& TEPS E42 SxYsu NDp.ol! itoTxN Rnvsu NnYsu ROSCLV Fxvn zc3ioP E43 ZNUCNP Rzsora Z44 F&su Im UP E4S DRES DRZS _- RDXDK Fxw- iiosxl- Ex6- ?3m= :eexz Z47 TXB66 WORK3 E4 n !X3DS– E49 =N. =% n. I=N= Tlo184 ESO R-UP PHIP PHIP D61DS EPS>4 ESi ua- DSM EPS34 0s0s E52. VB DVOL------I Es] WB EPS84 PXI DISSIP E54 ------DISSIP ~x= E55 ------Nm RTzR3u”” VOLE RTZR31S ES 6 VOLB ------s E57 RXOVOLL RROVOLL i58– ------t ~~~= ------I E60– ------E61 ------TKETIL E62 ------EPSTIL E63 ------TKEN E64 ------EPSN

Fig. 12”. continued 67 1. The Piston Face. Given the dimensions of the desired piston geometry, the user lays out a half cross section on graph paper, as shown in Fig. 13. At this point, one must decide on the level of resolution available and, based on it, define grid points along the piston silhouette, starting at the bowl axis and ending at the cylinder wall. The outline is required to follow cell edges in the logical mesh and not cut diagonally across a cell. With these points defined, it is a simple matter to create the input table. For each point, the generator requires the logical coordinates (i and k) and the physical coordinates (r and Z) in cm, relative to z = Obeing the lowest z-coordinate. In the input data, NPO is the number of points, and the coordinates are in the arrays IPO, KPO, RPO, and ZPO. In addition, NUNIF specifies how many zones out from the centerline are to remain uniform in r from the bottom of the mesh to the top (NUNIF z O). This feature allows one to maintain the best possible resolution in the spray region. Figure 14 lists the tabular information associated with Fig. 13. The mesh generator first assigns the vertices that have been specified in the table, then places remaining vertices, those within the fluid region, at average positions of their neighbors. The averaging in the code uses equal-weight springs. If this is found to create a less-than-optimum grid, it is a simple matter to modify the algorithm to use unequal-weight springs to shrink or expand cells selectively in the piston bowl.

0 0 1 2 3 4 5 r (cm)

Fig. 13. Chamfered bowl piston silhouette showing grid points to be used by the automatic mesh generator. 68 NPO 22 NLIJNIF c) 11 0.o- 0-.0- 2 i 0.28125 ~--0.0 ~. 3 i- 0.562?i 4 1 0.84375 0.0 5 1 1.125 0.0- 0.0 0.03 n“.,,k~4-C 0.275 0.5 Fig:M. (t’alml arinputda tacorresponding 0.90. 1.35 to Fig. i3: !,8 2, 375 2.55 ~.. ~.5- 3.3 10 9 3.4875 3.3 11 9 3.975 3.3 3.3 3.3 ~.~.

Note that up to this point, we have been discussing.the geometry ina-pnrdy twcE dirnensiimake.nse. In.fid, K.IVA-ILinitiallysets up only the j =- 1-a-zimuthalpkmei treating. the mesh generation a-s-two-dimensional, Theni for-cylindrical meshes, theggrr- erator-sirnply rotate~thisj = -1 confi~ation.do..ut.the.=is.t,a.c.reate the rwnaifiifig azimuthal planes, j = 2t?m-oughj = lNYP: Figure-15-shows-the axisymmetric mesh that results after rotation through 360°. Deactivated vertices lying entirely within the piston are not drawn. For a 2-D geometry with cylindrical symmetry, only the j = 2 plane is create.d ~yt,he rotation process= Some piston&ti~s-have-an-off set-bowl in the piston face, not concentric with the axis of”the cyiinder. ‘This is common iti diesei engines and represents a truly three- dimensional geometry. Subroutine SETUP automatically allows this option, through the -useof the quantity OFFSET in”the input ciata. If-OFFSET-= O~O;the result is.an axiiym: metric geometry? .asin-the.example.j ust=discmxsed..If CWFS13.T.tiO .0, however, the bowl -is- offset in the x-direction by thedistarw specified by the value ofOFFSS-T. An example is shown in Fig. 16a, which shows a bowl offset – 0.546 cm from the cylinder midline. To setup this geometry, the user starts by pretending it is an axisymmetric confi=gwation, as in Fig.. 16b, ~d.supphefi.~~e..t~h~l~r.i~~~ti~~~.ba ..-*As- .nn.... .+be~-.a~.. justed dimensions. T%e axisynunetric layout-is shown ifi Fig. IT and-the. asso-cMaM.ata ~di&ed3n.~_g. .1-8, .SETUP-first creates a @din the same=~na-~~~~=-as~n-the-previous exampl~ then cheeks the value of~l%%%%. Because--0 l?FSET =- — 0.546, the bowl and allvertices above it are displaced”to the left (negative x-direction) by this amount. The _radiallines are then straightened-from the new center out to theeyiinder wall. I%extj x- and y-coordinates between the bowi-lip and the cylinder wall are uniformly. distributed. Figure 19 shows three views of the final mesh SIiUIIP creates.

69 ,,,,,,,,,, ,,,, ,,,,,, !,,,, ,!, !,,,,,,,,,,,,,,,,,,,!!!

Fig. 15. The KIVA-11 computing mesh created using the tabular data of Fig. 14. The top view is a cross section through the mesh; at the bottom is a perspective view of the grid, in which the fluid region is outlined. NX = 13, NY = 16, and NZ = 16.

,,,,,,,

Fig. 16. (a) A piston cup with a bowl offset 0.546 cm to the left. (b) The user pretends that the desired configuration is axi-

II symrnetric and supplies the offset as a separate b parameter.

,,, ,,, ,,,,, ,, ,,,,,,,, ,,,,,,,,,,,,,,,,,1,,,

70 2

‘i

NPcl 28 NUN I F 0. 0.0 0.8 D. i 25 0-. !57-5 0.25. 0.425 0.375 0.325 0.5 ~_= o_. 22 O;-!= 0:75 0.09” 0.875 0.05 :.o~. ~ ; ~3- 1.125 0.01 1 ,S 0.0 1.5 0.02 1.75- 0.1 2.0 0.. 225_ 2“. 25- v. 43= 2.4-75 0.75 2.563 1.0 2.604 1.35 2.836 1.35 3.068 1.35 3.301 1.35 ~:. ~ ~ :’.s 3. m 1.35 ~> CyJ& 4;.~+ 4%23Q ;..Z 4. 463_ 1,-35 4.695 1.35 4-. SEz& i.35

Fig: 18. Tabular input data corresponding to Fi~.lJ’.

71 ,,

Fig. 19. Perspective, overhead, and cross-sectional views of the KIVA-11 computing mesh created from the tabular data of Fig. 15, with the bowl subsequently offset. The mesh dimensions are NX = 20, NY = 24, and NZ = 20.

In the previous examples, the piston cups have been round, as are indeed most cup designs in common use. Some designs, however, employ a square cup, the purpose being to enhance turbulent mixing. A square cup option is included in the KIVA-11 mesh generator and requires a 90° sector geometry (JSECTR = 1, CYL = 1.0, THSECT = 90.0, and OFFSET = 0.0). The example shown in Fig. 20 has NX = 13, NY = 12, and NZ = 20. It was created using the input data listed in Fig. 21, which shows three additional variables: . SQUARE = 1.0 indicates the square cup; SQUARE = O.0 for the round cups of the previous examples. . RCORNR is the radius of the corner in cm; at present we require RCORNR >0. . NSTRT is the number of zones with straight sides before the corner radius begins. A relationship between NY and NSTRT is implied, and the code checks to ensure that NY – (2* NSTRT) z 1. The mesh generator has been successfully applied to a wide variety of bowl designs. In addition to the chamfered, Mexican-hat, and square bowls of the above examples, we have modeled deep curved and reentrant bowls. The bowl volume is printed by SETUP as a check for the user, along with the total mesh volume. At the other extreme, a flat-topped piston is obtained by defining IWO = 1 and ZPO = 0.0 for all tabular points. For a fixed Eulerian grid, input ATDC = – 180°, SQUE3H = 0.0, and STROKE equal to the desired mesh height. (The result is the same with ATDC = 0°, STROKE = 0.0, and SQUISH equal to the mesh height.) Unless 72 ,,

,,,,,,

Fig. m Overhead~ front, and perspective view~-of-a-sqgare-cup mesh. ~ne mesh dimensions are NX = 13, ~~-?Vx~--_— ~~; ~1*~+~--=~~:

NPO 24 NUN I F o 1 0.. Oreo. o_.CKX3Q. ; 1 0.3442 0. 0000“ 3 1 0.6883 c .0000 ~. ~ j-..~~~~- 0. Wof.f. 5 1 1.3767 0.0000 ~ j j ,72QB 0.0000 7 1 2.9650 0.0000 7 2 Z Q650.. Q.. 3Qm- 7 3 2.0650 0.6160 7 4 2.0650 0.9240 7 5 2.0650 1.2320 7 ~ ~.Q650 i .5400 7 7 ~.. 0650 ! . e4E$o_ 7 e 2.0650 2.1560 7 Q ~ . Q550 2-. 4640 Fig. 21. ‘I!aMax.input.data.for. the-square- 7 10 2.0650 2.7?20 cup example of Fig. 20. 7 1 1 Z-AQE5Q. 24 OfQ)_ 8- I I 2-.27-15- 3.0800 9- 11 2“. 47%0 3.0800 10. 11 2_u_l@_ -3_ clsocl 11 It 3.3040 3.0800 ?2 ~: 2-.7t7o 3 i 0s00 13 11 3.9235 3.0800 14 11 4.1300 3.0800 modified by the user, the generator will create uniform SZ’S. The result is a simple cylinder, which is also useful for other purposes besides engine applications. The piston motion may be turned off to maintain the fixed grid simply by setting RPM = O.0 in the input data. This will also automatically turn off the swirl, because the swirl ratio is defined as the ratio of air r.p.m. to crankshaft r.p.m. If a swirl is desired in an RPM = 0.0 case, the user will have to patch the ANGVEL statement in subroutine SETUP. A nonzero value of OFFSET can be used with a flat-topped piston if one wishes to place the axis of rotation to one side of the axis of the cylinder. Thus, all planes would have an appearance similar to that illustrated in the overhead view of Fig. 19. The simple plane coordinates (CYL = 0.0) meshes shown in Fig. 9 are also created automatically. In 2-D, the j = 2 plane is identical to the j = 1 plane, but is at depth Sy (DY in the input data) behind the j = 1 (y = 0.0) plane. In 3-D, the j planes behind the j = 1 plane have uniform tiy(DY in the input data). This may be easily overridden if nonuni- form tiy’s are desired. 2. The Cylinder Head. Analogous to the definition of a piston silhouette, the generator uses tabular input data when the cylinder head is not perfectly flat. In the input data, NHO is the number of points in the head outline data, again starting at the axis and ending at the cylinder wall. The limiting case is, of course, the flat head, as in the previous examples. Here the use specifies NHO = O, with no further mesh generation data required after the NHO line. For a nonflat head, NHO >0, and the coordinates are in the arrays IHO, KHO, RHO, and ZHO. ZHO is relative to a value of zero at the lowest point. The head shape, typically a dome, maybe cylindrically symmetric, ellipsoidal, or semi-ellipsoidal when viewed from above. In addition, the dome maybe offset, again using OFFSET in the input data. If NHO >0, the head is offset rather than the piston bowl, if a bowl exists. An example that uses all of these features is the mesh shown in Fig. 22. Let us examine the sequence of steps in its creation. First, the piston silhouette is defined, as discussed in the previous section. This is a simple cylindrically symmetric shape with a slightly arched top, as illustrated in the bottom of Fig. 23. The first set of 11 tabular lines in Fig. 24 provide the definition. If we were done, the mesh would appear as shown in Fig. 25a. Second, the basic head shape supplied to the code neglects for the moment offset and ellipsoidal adjustments. This starting profile, drawn at the top of Fig. 23, and entered as the set of 17 lines following NHO in Fig. 24, modifies the mesh to the appearance shown in Fig. 25b. Third, the head is offset 1.20 cm to the left, specified by OFFSET = – 1.20 in the input file (Fig. 25c). Finally, the head dome is made semi-ellipsoidal. Whenever NHO >0, NEO must be specified. If NEO = O,no tabular data follow, and the head definition would be complete. Otherwise, NEO is the number of ellipsoidal or semi-ellipsoidal

74 k-pianes in the dome. The four columns-of tabular. infcmmatian.that.fcdlow. MEO-.are-the. tirays.lSC.~R;~-X;.SE~MJ~ and EH3MINIFKlNCX-VtKisthe index-of the correspond-- ing line in the NHO table. For a perfect ellipse both left and right; the value of IEMAX.is zero, using the semi-major (SEMINJ) and semi-minor (SEMIMN) axes specified at that ievei: Points along the.ellipse.are. determinexLhy. their. intersection. with.eachradial~ -id line in turn, which origjnate at the ellipse center. If IEMAX = 1,.as in this example, the. co-de again creates a perfect eilipse on the left, but on the right chooses the shorter of”the ellipse distance and a eirc-’ular arc ‘.vhos~radius-is-theRxHO -atthatlk-ievel. ‘This ensures that the ellipsoid on the left will smoothly join the circular arc on the right, as evident in the overhead”view of Fig. 22. The final mesh is also shown in-the m=-timofl-Fig..Md.

Fig. 22. Perspective views o~a K1l-A-H”mesh with a domed head: An overhead view is drown at.the lowerleft, aml a c-ut with pkmesj ~~= 11 through j = 16 removed at the lower right. The mesh dimensions are NX = 10, NY = 16, andNZ= I-6..

75 5

0 o 1 2 3 4 5 2

1 N

0 0 1 2 3 4 5

r (cm)

Fig. 23. Grid points used by the mesh generator for the mesh of Fig. 22.

F. Cell and Vertex Flags In many engine applications, of which the examples in the preceding section are typical, a number of cells are deactivated, as they lie entirely within the piston or the head. In order that calculational DO loops may easily recognize such ‘fobstacle” cells, in addition to the ‘tghost” cells in the i = NXP, j = NYP, and k = NZP planes, we use a cell flagging scheme. Cells with F = 0.0 are deactivated, whereas cells with F = 1.0 partici- pate as fluid cells. In a vectorized DO loop over all cells (e.g., Do 1014 = 1, IJKVEC), F(14) is used as a coefficient, thus permitting vectorization, as no testing is required. SETUP also defines a set of vertex flags FV. Analogous to the cell flags, FV = 0.0 describes a vertex lying entirely within an obstacle, and FV >0.0 describes a fluid vertex. Used in the automatic mesh generator and retained for use thereafter are unique flags for each of the following possible cases: FV = FLFACE = 1.0 for all vertices on the piston face, FV = FLBOWL = 2.0 for bowl vertices not on the piston face,

76 *1 o 0:0 0.86 0.5 0.84 %-.Q 0:87 0.77 ;:: 0.71 ~,~ ~ ~&-. 3.ct 0.54 3.5. 0.42 4.0 D. 29 4.5 0.16 5.0 0.0

::0 ~,o~ ~,~~ e . 3-3- (-). ~? 4.00 q. ~a. 3—, 95 1.20 3.90 1.50 3.75 1.8s2 ~.~~ 2.20 3.00 2.55 2.50 ~: ~5- 2:00 2.55. i .32- 2:63: 0..64. 3.18 0.54 3.66 0.42 4.05. 0.29. 4.50 0. ?6 5.0 0.0 i4- 0.33 0.33 0.67 0.67 0.94 0.84 1.20 1.00 1.50 1.20 %.80 1.50 2.20 1.80 2.48 1.95 2.68 2.00 2,93 2.00 ~., ~o: 3.i6

Fig. 24. Tabular input data for the mesh of Fig. 22.

-, - -,-- FV = l?hQbH = S.0 for all-of”sqgisfi region above the piston, FV = FLDONIE = 4.O.for.vm~icesuwithin the head.volume butnoton the head, a-rid FV = F-LIEEA-D’= 5:0 fonetiices-on-the- head surface itself. In a vectorized-DO- loop over all vertices .(e.g., D-O 1014- = 1, IJKALL), CV-MG- statements are used “that embody FV-information in such a way as to ensure that deactivated vertices have ml effed. .

G. Fuel Sprays

T-he fuel spray injection model in ~WrA-~is Siifficientiy ~e”lm~d] “~b~:a- wicie.variet.y of engine injectors or continuous sprays maybe specified through input data alone. For many users, subroutine INJECT should require no modification. Features of the injection ~mod&l=i~mi-u&-a-multiple-ormultihole nozzle capability, with continuous or puised hollow cone or solid cone sprays whose origin, profile, and orientation are easily specified. Pulsed sprays ray be sinusxkla-1; ~qu~r~~+ave; or-be swpplied -witha tabula~-~elocit y-~ofiie- a~ptiate-for=hoi e-type nozzle injectors. Either a fixed “particie radius or a distribution of~adiimay. heinjec.ted..

77 ,,,,,,,,,

a) b)

,,,!!,,,,,,,,,,,

c) d)

Fig.25. The fourstages inthegeneration of themeshin Fig.22: (a) The original mesh with an arched piston top, (b) The addition of a domed head, (c) The offset of the domed head, (d) The adjustment of the head shape to a semi-ellipsoidal form, when viewed from above.

These features are now discussed in detail. 1. Spray Oritin, Profile, and Orientation. (See Fig. 26.) The input quantity NUMNOZ specifies the number of spray nozzles (1 to 12). For each nozzle, the radius in cm to its location is specified by DRNOZ, measured from (XO,YO), which is the cylinder axis, or in the case of a planar mesh, the front-left-bottom corner of the mesh. The azimuthal rotation in degrees is given by DTHNOZ, measured counterclockwise from the y = Oline. DZNOZ, in cm, locates the nozzle in the axial direction. If DZNOZ <0, it is interpreted as a distance below the topmost point of the head. This is generally appropriate for engine applications. IF DZNOZ >0, it is interpreted as a distance above Z(l). This is the appropriate choice for a spray in a fixed mesh, such as a spray combustor or burner. In order for momentum exchanges with vertices to take place properly, subroutine INJECT ensures that the spray is at least one half cell out radially from the axis in sector

78 -—I * x. -Dzw \I I /“ \ I I I !. .

_TH.Tx2 “\x z

~. Y ~./“

I XY I

l?ig: 26. The input quantities DRNOZ, DTHNOZj and D-ZNOZiocate each spray nozzle; TIUIXY and TIITXZ-define the spray axis in 3-D-space; C“ONE and DCONE ail~lv=for=hollaw cone or-solid %mesqways meshes-a-rid either at least a half mll below=the topmost-poird-ofthe head or a iiaifcell above Z“(1). CONE, 13CONE, TILTXZ, and TILTXY, all supplied in degrees, define the profile and orientation of each spray jet. CONE and DCONE provide either a hollow cone spray or a solid cone or pencil spray. C.ONE is-the.meanmmeangle for hollow-~cne Spraysj and DCO.NEis.fte thick-ness of the spray. If CON-E ii+input identically equai to DC!ONE, a sol.ickpray results. “Theorientation of the-spray axis fiireaeh nozzle is defined by TILT-XZ and’HLTXY, where TIL’ITXZ gives the x-direction inclination from vertical in the x-z plane and TWTXY- specifi~=stire-mtation-ofi”he-spray axis in the x-y pl”ane. A-swit-hlYI133WUZ~ ‘TH-f17X-Yis .measuredcoMterc16 ck-wi&e~rom the positive x-axis.. In the ease-of’ a-seetor mesh with .NY =13 the model assumes- thatthe spray jetis azirnuthally -centered-in the half-degree sector; by enforcing TILTXY = THS13CYT/2: x_ !$EyI!3Qw13efimiti-a~-. Sev~ral input quan tities~-baraetw~zethe-k~o;w The quantity PULSE differentiates between [email protected]. injection. PULSE = 7+ 0.0 is used for a continuous spray, for which TSPMAS is the mass flow rate in g/s, and TNPARC is the number of computational particles injected per second. Alternatively, PULSE >0 defines a pulsed spray, for which TSPM-AS is the total mass in grams to be injected, and TNPARC is the total number of computational particles to be injected. Clearly, the accuracy of the spray model improves as the number of computational particles is increased, but the code can be significantly slowed down, especially by the particle collision subroutine. The code automatically calculates the mass per computational particle to ensure that the correct total mass or mass flow rate is injected regardless of the choice for TNPARC, which is governed solely by computer time and storage constraints. Some typical values for TNPARC that we have used in engine calculations are 500-1000 (2-D) and 2000-5000 (3-D). For a continuous spray, we typically inject TNPARC = 40000/s. Because of evaporation and an outflow boundary, particles are continuously being destroyed, so that we can get by with a dimension NPAR of only 5000 for the particle storage arrays. Computational particles are injected with speed VELINJ cm/s, and the angular distribution of particle velocities is uniform within the internal DCONE. The density of the fuel in g/cc is supplied as RHOP, and TPI is the fuel temperature in Kelvin. Particles are moved in subroutine PMOVTV, which will also add the effects of turbulent velocity fluctuations ifTURB = 1.0. Three types of pulsed sprays are available: PULSE = 1.0 defines a spray whose mass is injected in a single half sine-wave pulse. PULSE = 2.0 defines a spray whose mass is injected in a single square-wave pulse. PULSE = 3.0 defines a pulse whose velocity profile is supplied by VELINJ, which here is a table of NUMVEL (up to 100) entries. For this case, subroutine RINPUT calculates the total fuel mass predicted by the velocity table, using the sums of the nozzle areas ANOZ( ), and corrects the velocity profile up or down by the ratio of mass desired to mass predicted. This case is appropriate for hole-type nozzle injectors, rather than pintle nozzles, as the nozzle areas at present are assumed to remain constant with time. For the cases PULSE =0.0, 1.0, or 2.0, however, NUMVEL should be 1, the velocity being assumed constant at VELINJ(l),. (A comment in subroutine INJECT lists a one- line modification to allow the sinusoidal case to have a sinusoidal velocity profile, in addition to the mass profile.) In addition, ANOZ( ) can be input simply as 1.0 for these three cases. Injection commences at crank angle CAIINJ and has a duration of CADINJ degrees. CAIINJ is given in degrees ATDC, and hence usually has a negative value. Injection may also be controlled in terms of problem time, which is more convenient for continuous sprays and other nonengine applications. In this case, the appropriate starting time and dura-

80 tion are supplied as TIINJ and TDINJ. For a continuous spray, TDINJ should be set to m. A-negative value for TIINJ-inciicates to the code that CAIINiTand-CADINJ are to be used instead; 3: Par&4e-Radius. E-ith-eradistrfuutio~rof-patiicle- radii (INJ131ST-= I) or particles-of a fixed radius (INJDIST = 0) maybe injected. ‘When a drop size distribution is spacified, we sample randmnly from a--dis~ribation abeutthe-Sautermean-radiusiS?tiR), inputin-crm T-hedistltiktiom-is –pattemed after experimentally o’bserved “data and “is described in AppendixJl. In the c-aseo&FIxed raditisj tlievaltie supplied:i-mder-the nme<iRisinteW_reted as the fixed nozzle radius, in cm. The aerodynamic breakup model shoultie.~-wd. (BREAKU.P = 1.0) .to.crAaa~&rum-of.tim%[email protected]~..kreskup .nmdd.&appsops:- ate in general for both the INJDIST = 1 and INJDIST = Ooptions. AMTO is the initial amplitude of droplet oscillation at the injector, based upon a Weber-numbex.eetimate. After injection, the particle size is.reducecWy-break~ M3113EAKIJP = 1-.0,and through evaporation, if EVAPP = 1.0. Liquid fuel particles disappear as they. evaporate, as sukirouti-ne REIPACKdestroys any. particle~who~-mass falls below- 10-3 of.the .massaf an injected particle. Conversely, particle size increases through coalescence from collisions, if IH31.iI13E-==1. Particles-are splitin REPA-CK into two identitai particles, each with fiaifthe number of’droplets, if”tlieir mass grows to twice the mass of an injected particle. 4. Stochastic In.iection. When using the large timesteps possible in K.IVA-11, the tendency iS to inject in bursts, resuiting”in discrete clumps ofiomputational particles. T=heseclumps-ma-y=then mav~ma~e -than o~e-e~ll pereyc}e; causing-armmevmr coupiir~-g- with the mesh cells and vertices along the particle path. ‘Ihnitigaiat.hissource of.computational inaccuracy, .KW.4-11 injects-eswh particle at some random point along the particle trajectory behind the injector. The particles are then immediately moved forward in subroutine PMOVTV to.their.effective initial hma. tions. This stochastic injection offers a smoother and more uniform particle distribution,

resulting in .improveci “coupling- wit-hthe-mesh j and better- statistiVw‘Whefi=spiay.---Wdrticle positions and radii are averaged over time.

H. Spark Ignition Spark i~ition is provided by a special energy depositionat.the .end-of.the kinetic chemistry mI’broutirm-CHEMT Tiie i~ition window is specified either by crank angle (CAIIGN to C.ZWIGN-+ C“ADIGN) or problem time (TTIGN- to TIIG-N + TDIG-N), “in a manner analogous-to th~=pedieation of the injeetion -window discussed-above: During ignitio~ the spedlc internal emxgy in *&.e-specified ign~tio~.cell(s) is increased by a factor of (1.0 + XIGNIT*DT) on.each.timestep. If-the temperature in the ignition cell(s) ~~ reaches 1600 K before the end of the ignition window, as it ordinarily does, then the special energy deposition is terminated at that point. The ignition cell(s) are specified in the input data as the DO-loop range defined by llGNL(l) to llGNR(l), J_IGNF(l) to JIGND(l), and KIGNB(l) to KIGNT(l). This allows the choice of a single cell or a block of cells. If JIGNF(l) = 1 and J_IGND(l) = NY, ring ignition will result in a 3-D cylindrical run. Dual ignition is an optional feature, as a second ignition region may be specified by setting IIGNL(2), IIGNR(2), JIGNF(2), JIGND(2), KIGNB(2), and KIGNT(2) to nonzero values. Spark ignition in our UPS-292-SC engine calculations was accomplished in a unique manner. In the UPS engine, a pencil spray of fuel impinges on a spark plug which commences firing when the spray starts and continues firing for 35 crank angle degrees. Our procedure was to not allow fuel and oxygen to coexist in the cells containing the spark plug tip during the 35° period. Each cycle, any available fuel or oxygen was consumed through direct conversion to COZ and HzO in accordance with the oxidation reaction, with appropriate heat release. While quite satisfactory, this procedure is applicable only in such specialized circumstances and would be totally inappropriate in more general applications in which the fuel has experienced significant evaporation and premixing prior to ignition.

I. Initial Bessel Function Swirl Profile Internal combustion engines are designed to impart a significant amount of swirl in the incoming air, to aid in turbulent mixing and enhance combustion efficiency. The simplest model assumes that the swirl velocity has a wheel-flow profile, but this is not usually a realistic assumption, as the turbulent wall boundary layer forces the swirl velocity to decrease in the wall region. From experimental observation, modelers have determined that a Bessel function profile more accurately represents the flow. Figure 27 illustrates the Bessel function velocity profile provided in KIVA-11 and compares it with wheel flow for the same swirl number. The quantity a (input as SWIPRO) is a dimensionless constant that defines the initial azimuthal velocity profile and lies between 0.0 (the wheel flow limit) and 3.83 (zero velocity at the wall). A value suggested by Wahiduzzaman and Ferguson 58 for typical engine applications is about 3.11. We define our Bessel function profile to give the same angular momentum as wheel flow with the same swirl number. Thus the initial slope of the a = 3.11 curve is necessarily higher than the corresponding slope for wheel flow. A second input quantity, SWIRL, is the initial swirl ratio of air r.p.m. to crankshaft r.p.m. When viewed from above, the swirl is clockwise if SWIRL >0 and counterclockwise if SWIRL <0.

82 h ;. :. :. :. :. :. :. ,- 1- V=Z513 a=3.11 (~4339)= –—. ,., I / \ i ,a,,..’.’ j.s!A!iRL= ‘- ..: ,-/ ~~~-si~o~ I. /- ,..,’><

,:.-/ \ ~~$o+.:’’””.’.’”. \ / *?.+!,:,:’”’.. q,,,:=. ... / ,.,. ,., ,/ ... 1.

/ ./’. ,.,.&;3;3--.\_ (vm=762j” “\ .Y’ \ ./ \% ../! \

V=O.90 O+Q M- 0’2 03 M- cm 0.6- o.?- 0.8- CM- !.0 #R=

Fig. 2’7. B-essel-function swirl velocity profile provided in KIVA-11 setup,

J. Fuel Library “Tilerelease version o~KIIVA-H”assumes that 12-chemical-species are present, although .thi~ number can .be increased Or deerea-sed as-needed. T%e12speeies, appropriate ifma varie~-ufmmbustionapplicatiolm; arc-l--=fuel, 2 = 02, 3“= N2, 4- = C-G-2,5 = E2GY 6=-H- Y = H7, 8 = c); 9 = It, io. = OH; _W= C33.andIZ= ND-..T.he sample input data deck=c;ntains kinetic and equilibrium e-hemis~ry data-that correspond to these- Uspeeies. Enthalpy tables are required for all species, and because species 2-12 have been defined, enthalpies for them are provideiLhdlATTA&a@nents in.subroutine IUNPUT.. What remains is to define the fuel, specie~.1~ ., which requires a-number ofother- prop- erties in addition to its enthalpy. For this purpose, KIVA-11 contains a library of the m... a ~.. — . thermop.hysicai’plwpel%i~s-of~~’c~-on- hydro-carbon fuek, emboded in J31AU~K KJ-ll”-A FUELIB. At present, the 12 fuel choices are methane (CH~), propane (C3~s), n-heptane (C7H16), n-octane (C8H18), n-dodecane (C12H2G), ~=$~:~~.~fi~ (~ ~&f2x); n-tetradecane (C14H30), n-hexadecane (C16H34),

83 acetylene (C2H2), ethylene (C2H4), benzene (C6H6), and diesel fuel (DF2).

To select from this list, the user supplies the fuel formula as a mnemonic in the input data. Subroutine RINPUT then calls FUEL, which performs the following steps: 1) The fuel formula is correlated with an entry in a tabular set of mnemonics, to verify that the requested fuel is one of the 12 available and simultaneously to obtain an index for accessing the data associated with that particular fuel. 2) The fuel enthalpy table is loaded. 3) The molecular weight, critical temperature, and heat of formation are defined for the fuel. 4) If the fuel is normally gaseous (methane, propane, acetylene, or ethylene), the input flags for parcel evaporation, collisions, and breakup (EVAPP, KOLIDE, and BREAKUP) are checked. If they are all off, FUEL has completed its task and control returns to RINPUT. If any of the three flags is on, the run is terminated with a mes- sage that the input data are inconsistent. 5 If the fuel is a liquid, several more quantities are set. These are the tables of liquid latent heat of vaporization and liquid vapor pressure, for use in droplet evaporation, along with a table of liquid viscosity, for use in droplet breakup. In addition, the slope and y-intercept of the surface tension vs temperature linear fit are defined. These are based on the surface tension at 350 K, which is a typical fuel temperature, and the critical temperature of the fuel. These are used in the surface tension linear fits in droplet collisions and breakup. Finally, the coefficient for fuel diffusivity in air is set, for use in droplet evaporation. The tabular information in FUELIB has been drawn from a number of sources.ss~sl-ss~sgIn several cases, these sources do not contain suffkient data to complete the tables at one end or the other of the temperature range. This is true at the low temperature end for some of the latent heat tables, and at the high temperature end for some of the liquid viscosity tables. The comment lines in FUELIB identify the sources of the various tabular data and note where extrapolations from published data were made in order to complete the tables. The user should thus be aware that development of the library for less-completely documented fuels is an ongoing project. Contributions would be welcomed for inclusion in future code versions, both for improving the data for the existing 12 fuels and for fuels not presently included.

84 Thefinal consideration concerning fuel”is its inciusion in.akine-tic.chemical reaction. To accomplish this, the user must supply an appropriate input data set; as required for each kinetic reaction. The data set is comprised of forward and backward pre-exponential fac.tm~,.activation temperatures and temperature expommts~ akmg with stoiehiometric species coefT12ients on the left and=right sides of the reaction and exponents of species mneentraiion -in-bath-the-forward -and-baekwa~d-~-=tes-o-fthe-reattion. Kinetic chemistry idismsse-d:in -detail in .Appendix-I;

K“. Inrlow and outflow Boundaries Some engine and rmnengine applications-have geometries thatrequire useof-irdlow- and outflow bmmdaries, Acwrdingj:y, ‘we-l~ave-provided-in~low and-out~lo.w-o~~~n-s.me-a- type that may be.encounterexL. ~Input..~ag.B.O-T.IN-= LO, .tlm entire bottom !mundary of the mesh is an inflow boundary. If BOTIN = 0.0, then a rigid wall is used for the bottom boundary. At the inflow boundary the normal velocity WIN is specified. Either one, or both, of”the ri~ht and.to.p tmun&u&s oiltke.mesh maybe outflow- boundaries. Input flag RTOUT controls the right boundary specification, -with.RTOUT = 1..0denotingoutflow and RTOtTT= 0.0 denoting a rigid wail. ‘The boundary lies between the I = NX and 1= NXP-planes of”cells. Input fIag TOPOUT performs an analogous role for the top boundary, ‘whiell is the imundlwy betwe-emthe-K-—-lti~’and-.~= ~Y27Pp~anes. A-toutflow boundaries, the pressure is speci~~ed “to‘be 1npul v.ahe F51h!H3. A!t kth. lWIUnd-a-rkSj we assIwme the flbw.is subsonic. We now-descr~be the inflow and outflow options ii-detail; telhvhat- subroutines-are modified -t-ail~eu~umte-them; amd-tell how they may be modified “for superson ic-~fbva. A-lthm@tie._~hw..am.&~.u.tllaw.o .-~nthn~..--.%. -->a w –-r.a.th-L ..-.Q.R1..in-.iid.... “W.A,.i~.;S..he~~ ~~at ~~eycan be used ~s-a ~ide-t~ in~~rporating-. boundaries -withdifferentflocatic}ns-amd-. different conditions. in.additiim to the-normd.velcxity WIN j at in flowboundaries -wespeeify referene~ speeies-,mass densities SPDINTO(M))specific turbulent kinetic energy TKEAM.B, and turbulence length scale SCLAMB ( - k3’2/e). The reference densities are at reference pressure EWNH3;-the valties thatare actually imposed at.thsinflow boundary are obtaine~crrm-

Ifyanlb SPDfN(M) = SPDINO(M) [P/PAMB) ,

where P is the computed pressur~.in.the.cell .with .-Kindex .equal to one immediately abov+ the inflow cell and Yambis the ratio of specific heats of the inflow mixture. Thus at an inflow. hrmndary. we are imposingthe species .mass-fractions-a-nd th~tropy of the.inmrning. fl-tichncLoM.aining. the .pressure-by-extr.apolation zmd the derwiticx-fram an isentropieg~s - equation of state. The inflow internal energies are obtained from pressure P and densities SPDIN(M) using the-equations-of state (10) and”(l13. One ififl-owtangentitil-velocity com- 85 ponent is specified and the other is calculated. In cyl ndrical mesh geometries, the azimuthal or swirl component is specified to be zero. In planar geometries, the y-component of the velocity is specified to be zero. We note here that the values SPDIN are used for computing mass fluxes but are not used to compute diffusive fluxes at the inflow. In fact, the diffusive fluxes of all cell- centered quantities are taken to be zero at the inflow boundary. This will introduce little error if the Peclet numbers, which give the ratios of the convective to diffusive fluxes, are greater than unity. If this condition is not satisfied, then the appropriate coding changes should be made to calculate diffusive fluxes at the inflow boundary. The above inflow boundary assumes subsonic flow. If the flow is supersonic, then all thermodynamic conditions and all three components of the velocity should be specified. At the outflow boundary, the pressure PAMB is specified a distance DISTAMB outside the outflow boundary. This is accomplished in the following manner. For a regular cell face a that lies on the outflow boundary, one of the faces y of cell-face control volume a also lies on the outflow boundary. In differencing Eq. (86) to find (uA)a}3, we take the pressure on cell face y to be

&/2 . PAMB + D[SY’AMB . [@ppB + (1 – @p)pnlLjk p=. . , fid2 + DISTA MI] where 6Z = V~k j IAal and cell (i,.j,k) is the interior cell containing face a. When DISTAMB = O, this gives p = PAMB. This is a true specified pressure condition, which unfortunately reflects acoustic waves perfectly and can affect the upstream flow in subsonic calculations. Taking DISTAMB to be a characteristic dimension of the computational region greatly reduces acoustic wave reflection at the outflow boundary. This also allows more rapid convergence when computing steady state flows, by reducing the problem time required to reach steady state. In addition to the above pressure specification, to im- plement an outflow boundary we set the vertex velocities equal to those one vertex in from the boundary: u~,j,~~}~ = u~,j,~~ at an outflow boundary across the top, and u~x~j,k = ujvx,j,k at an outflow boundary across the right. We also use donor cell differencing to compute outflow convective fluxes. When the flow at the outflow boundary is directed out of the computational region, no further specifications are needed. Occasionally, however, the pressure condition will generate velocities directed into the computational mesh. When thi#’dccurs, it is also necessary to prescribe the mass fractions, entropy, turbulent kinetic energy, and turbulent length scale of the incoming fluid. This is accomplished through input reference species densities SPDAMB(M), referenced to pressure PAMB. The inflow densities and internal

86 (energy are then found in a manner analogous to that used at the inflow boundary. The. infIow turbulence quantities are taken to be TKEA-MB and SCLA”MB. No nmdificzitiorrof-this-outfiowtxmndary treatment is required “when the flow IS siipersonic.- Although-sW-cification- of-the pressure is incorrect in this case, the errors incurred by this specification.wi llmcd+uqmgate upstream in -the supersonic flow. .Anumberofsubroutin~-aremodified to allow fortlw inflow anchutflow-options. Vertex and cell face.velo.cities .are.prescribed in subroutinesBCa-milMl&G... Sin~.-s isnot. specified near inflow or outflow bom-dariesusing Eq. (53), this specification mustbe-dis- abled in subroutines BCEPS and 13CR13SE. Similarly, law-of-the-wall velocity and temperature conditions siiouid’fie diisabled””insubrouti-ne LA-WALL. Convective fluxes are modifTed-in subroutine C-CFLUX~and the pressure is specified at the outflow boundary in subroutines PEXDIF and R13SP. In making any modifications for inflow or outflow b-mndnries; the-user shouktconsider tlie~.subrcmti-nes carefiuiTj-and use the existingcade asa.gy~ide.

d.1- Qu@Jt Monitor prints are produced on the user’s.remote.terminal jQur-unit 59) atleast

(e~e..y. 25th cycle. The variables -~J.4-Dprints-a~~listd in subroutines NETvV-CW aniF cp~~~~qy.I In ~d& tio~i in formatiol~a] -an~errormessages are always Sent to the USer’S terminal. Except for the above, all ‘KIVA-11 output-is wri tten .tofile PLC)T.,which can be scanned on a CRT-equipped terminal mderdisposed to the opera-ting system to be proc- essed”onto microfiche. A fbur-line monitor print is written on PLOT every. cycle, inaddi, ticm h-copies-of any messages sent to the remote terminal. Numerical cell data are opticma]-]y a-va;lable, being-producedo niyif.the-in~t.fl-ag llP*R = 1. The ‘v$%H% statementstin-subroutine LNGFTTI’ indicate the variabies listed; whiclh may. eas~1~.‘bech smge&. 13&eausemi.thevast~a~naunt=of:num’~ers-a .multi dimensional

,~mdepmduces$ we almost alwa-y-s-setLPR = Oand rely on-plots, excephvhen-d-ebugging. Even then, an interactive debugging utility is generally more useful than a blanket cell. pri~3*.bWawse=tiw-=d&i3itiy-c=n-print seieete& mir.nkrsrat the user% termi-nai’ at any iksireci point in the calculation. GiaphiEs-are-tfie- mostusefu] form of”out_p=ut,and “information for computer- generated’plbts % ako -written-to filii PHYIT. ‘TIIis-is-ibllowe dbya short summary of system totals, -computed by-subroutine-GLOBAL. S-ubrouti-ne FIJIXJUT is responsible for callin~the.variousgdot.and-print.submu- tines. Output is automatically. provided for cycles Oand 1, and thereafter at cyclic inter- vais (NCFIL-Ni).~time intervals (TWPLT), or crank angle fi.terv.al~------(CA FILM), as specified-. irrthe input data.

87 KIVA-11 can provide three different types of plots -- zone and spray, velocity vector, and contour. The first set of zone plots are simple two-dimensional views through (x-z) plane j = 1 and the opposite (or closest to opposite) j plane if 3-D. The first plot is a mesh cross section that shows all zone edges. The second plot is the mesh outline plus only those spray particles currently in the j = 1 plane and the opposite j plane if 3-D. The third plot is again the mesh outline, but with all spray particles plotted regardless of their j plane. A fourth zone plot is an overhead view of the mesh that also includes all the spray particles. If there are no spray particles present at the time the plots are drawn, the second and third plots are omitted, as is the fourth plot unless the output is for cycle O. For 2-D applications, only the first two zone plots described above are created, as the third and fourth plots would be meaningless. The remaining zone plots, in addition to all the velocity vector and contour plots, employ our perspective plot logic. For 2-D applications this is automated, but for 3-D applications the logic requires specification of input data. This is all described in Sec. IV.M. The plotting routines provided in KIVA-11 are adequate for the needs of many users. Because these routines require only graphics primitives (point plotting and vector drawing), they are adaptable to other systems without too much difficulty. Other users, however, will want to use KIVA-D with their own graphics post-processors. Accordingly, we have provided the basic connective linkage for post-processing in KIVA-11, which will simplify the task. It is controlled by the following three input quantities: ● IPOST is the post-processor flag, where Omeans no post-processor file is desired, 1 means to make post-processor dumps onto file TAPE9 starting with this run, and 2 means to continue dumping onto a previous TAPE9, which must exist in local file space. 9 CADUMY is the crank angle at which to start dumping onto TAPE9. ● DCADMT is the crank angle interval between dumps, after crank angle CADUMJ? has been reached. TAJ?E9 is initialized or resumed by subroutine DMI?INT; writing of file TAPE9 is performed by subroutine DMPOUT. Because each user of a post-processor has their own specific needs, the variables written in the DMPOUT routine provided in KIVA-11 are intended only as an example of those quantities that are generally useful.

M. Perspective Plots The development of three-dimensional computing techniques has brought particular difficulties in the effective presentation of results. We have attempted to display as much useful information as possible by offering perspective pictures, made in the same manner as a photographic record of a three-dimensional scene on a two-dimensional negative. The perspective plot logic described below is used in producing zone, velocity vector, and contour plots for both 2-D and 3-D applications. However, for 2-D applications,

88 fi~JA-~=~~bm-utineRTAINJT- autam~timlly-specifies simpk ha-d-cm HI yiews-fdr eadi quantity plotted; and the user needfiot3e-cmwerne&wi tli supp@ing perspec-tive view data. The plot-generating subroutines ZONPLT, VELPLT, PVPLOT, and CONTUR can create a variety of perspective views of the computing grid with spray particles, in addi- tiwrAwflui&velovity vector; spray parcei velocity vector, and “contour plots of selected-cell- vatiables; F-orthe grid=plots, some-selected “number of the three ‘bounding faces of our ~ti=Adw.Wid.are..to. he.&awn.. Em-the vec+ar.and wntour plotx+however, we shnply . . outhneth~ges-of the grdand presenka Slect&ld~pzc#Tf&t~rs-ar. cmnixmrs-wiiJ-&Ythis- framework. O-nly-a-few-simple concepts are required+~ deseribehow a-perspective view is gene~ ated; ‘The fundamental cu~me@fisth=A-af2he.transparent image plane, anaiog~us-to tb fihllima.~ .m.whiEM.he .pspectitie J&la@. -m= G. .t!zwed (-Fig. 282. ~.G__~a~.pL~~>;z*-&;*t*a- tion~ however., We eho~e-a-nirnag~pkme -perpendietilar-’~ the averagelimmf>tight; which . \ , ,— exten&fmn-some-point-( xC.,yc,zcJnear the mesn center out.to the eye point (Xe,ye,Zz), assmned.to.lie.we--lxmtside the mesh.

-4,...,, L . A,.., .——-—.. ..-% w— “- —., . ~.

.. M.Gik

Fig. 28. Cons-truetion ofperspective picture onan irnagg pkme~ From New Principles of Linear Perspective by Brook Taylor, published in 1715.

“89 A pair of perspective coordinates (e,q) relates a point on the image plane to a point (x~,y~,zo) in the mesh region and can be obtained by performing suitable transformations. In our case, these are

in which

x =(x —xc)cos o+(y-yc)sino ,

; =ccJs@[(y-yc)cos El –(x–zc) sinol–(z–zc) sin@ ,

; =Shz($[(y- yc)coso -(x- xc)sino l+(z-zc)cos(o .

The two angles 6 and@ measure the rotation and the tilt, respectively, of the line of sight with respect to (x,y,z) space as indicated in Fig. 29.

Fig. 29. Relationship of the rotation angles@ and 6 to the (x,y,z) coordinates.

90 From the figure, it isevident that

g= [(x e –XC)WYe –y)v,c

r= [[z —z)~+gy, e c

from which

if viewed “from the firont, (ye — yc) < O; Ifv” lewed from beliind~ .(ye —ye.) >0, then

m.1L.ae-Jltigln-amd.Sraie-Of-the (~,q) coordinates-are not .req.uire&bee~wfdmi3se a ~ ~mm%antshiftthateenters the inesh outii~~e--i~~t~l~-~ot~crame-and a-constantsca~e factor that maximizes the size of the mesh drawn in-the.wailaMqdQt.frame area. T-his scaling is done-in KIYA.11 by first computing the (~,q) coordinates of all mesh vertices and then testing fiirtl-ieir maximum and minimum values. Because a straight line segmentin-three. dimensional .space transforms into a straight line segment-in the perspective view, only the end points require transformation accmdi~~gtuthe-ahve-e~ations; and the resulting points are connected “by a straight ]ine. h .K2WA-11;+@’&laV&kfined xc = yc = ‘0;3”Yan&zC = “(zOf the ~yii-n~e-~l~a&minus- half+k--ssroke), assumingcode applications will “’beto internal “combustion engines. For 3~D-plme-coorfintie~ .the.l~sw.&auid.redefifie Xc,.yc,.andzc to lie at the center of the mesh. These sta-tements~-an beeasily changed ii KIVA-13subrcmtine RINPUT-. Current= ly, we allow up to five views each for grid plots; velocity vector plots; and contour plots, as the per:t.lie~paramete~ LV attbe begkning -code: Requi redi nput-qu antitiesfor eaeh individual 3=B=plot=a~w-asfbilmmx Grid Plot: XEZ, YEZ, ZEZ, IFACE(6), IEDG, in which the first three refer to (z,,y~,z~) in (x,y,z) space. IFACE is a set of six integer flags that correspond to the six bounding faces of the logical mesh: left, right, front, derriere, bottom, and top, respectively, where a value of 1 means to include and a value of Omeans to exclude the face in the plot. For the 3-D pseudo-polar mesh, appropriate perspective-view values are O, 1,0,0, 1, 1 to draw the right (cylinder wall), bottom (piston face), and top (cylinder head). Finally, IEDG = Owill draw line segments between all vertices lying on selected faces, and I13DG = 1 will draw only the outlines of the selected faces. The IEDG feature is intended for use in 3-D plane coordinates. Velocity Vector Plot: XEV, YEV, ZEV, ISLV, JSLV, KSLV. Again, the first three refer to (Xe,ye,Ze). The last three are integers of which only one can be nonzero. The nonzero choice identifies the I, J, or K index of some plane in the logical mesh where we make a “slice” and draw vectors of the vertex velocities as they would appear normalized in the plane. For an I slice [0 < ISLVs (NX + 1), JSLV = KSLV = O], the plane is in (y,z) space. In a 3-D pseudo-polar grid, an I slice plot would be a wraparound at some radius and probably would not be too useful. Of real value for this grid, however, are J and K slice plots. For a J slice [0 < JSLVs (NY + 1), ISLV = KSLV = O], the plane is in (x,z) space. In a 3-D pseudo-polar grid, the velocity vectors for thej plane JSLV are drawn on the right side of the frame, and vectors for the opposite (or closest to opposite) j plane are drawn on the left side of the frame. For a K slice [0 < KSLV ~ (NZ + 1), ISLV = JSLV = O], the plane is in (x,y) space. In all cases, the vector length drawn is scaled to the maximum velocity in the plane being plotted. Contour Plot: XEC, YEC, ZEC, ISLC, JSLC, KSLC. These are analogous to the quantities defined above for the velocity vector plots, except that individual contour plots are drawn for each view specified. Contour plots may be drawn for up to 26 different cell variables, selected according to the set of 26 binary flags in the ICONT input line. The DATA statement in subroutine CONTUR and the comments in subroutine FULOUT describe the sequence. The same remarks concerning slices apply to contour plots, but because contour plots relate to cell-centered variables rather than velocities, the ‘c+ 1“ on the range is inapplicable. As provided here, contour plots are composed of vector segments joining points of equal value and are linear in contour increment. Contours are automatically connected across the center (at I = 1) in both J and K slice views in the 3-D pseudo- polar grid, and are drawn to the j = 1 andj = NYP boundaries in K slice views of a 3-D sector mesh. In both velocity vector and contour plots, it is best to define the eye point such that the line of sight will be as perpendicular as possible to the selected plane. When the eye is

92 not ;ocated-too-far above the mesh, .amore pleasing .appearmce-iiofien..mo.duc~d.hy.setting.

Ze = Zc. As a result, @ = O,which keeps the image ~kme perpendicular to the (x,y) plane. Often we prefer to eliminate perspective entirely. For example, a set of (x,y) velocity vector and’contour plots at various k-plane ieveis viewed-from straight overhead generally proves more usefui”than views from some artistic angle, To produce a straight overhead . VIew, Sek-xg.-=-’ye = 0:0 ~ndz@ = ~(101‘ kadequatej. l+Oi~.hW.~Te~,.~fiat~e = 0.0 ~-iii cause -the3=13-peub=~mla~p.lotto-’~e rotated.45~ wh.thti-the-pwi odic.houndar~~-(J-= 1) will appear at the 4:30 position. To mentally orient ourselves, we prefer to always have

the periodic ~ound~ry at t.~~ 3 ~’ci~c~ position. This is achieved-autom-atica-lly if-we ill~tead ‘SPe~ifYye = – RP(.XNI?O>rather ttian ye = 0.0. The eaordina%~~ystem on our CRT face has its cn+gim(OyO)atfihe-upper ieft-corner, m.&the-values-ofthe two raster indices increase to the.ri~t-tiwn.ta. maximum values of”l”02-4..A conversion from image- plane. cmndinates t~.ccwrespcmding CRT mordinates-is - thus-required for all plotting. In KIIWA=II;the view ‘on-the-ima-~eplane extends from EL.to 5fi_and from q~ 10 qT, with CRT counterparts-FGW to FIXR and31Y”B.to.lTYT. To leave ro~rnfbr-lfidin~ .welimit.the available CRT-face to 1022 ras%erpoints-wide -and 900-

- .-s. -. ,. points high. ~lthm this rectangular region, we maximize the image-plane view drawn, which requires the ratio

If the image-plane view is higher than it is.wide-[thati~~ll <.(1022/900)], then the CRT plot range is given by ~~z~.~= ~~.1 _ 450(X”-D”), FIXR = 511 + 450(XD) , FIYB = 900 ; and FIYT-= u- .

In-either”case, we can now calcuiate the two conversion factors x~o~: = (EIXR —..-=-m~~)/(&R _~lA; 93 and YCONV = (FIYB – FIYT)/(q~ – q~) , required to translate image-plane coordinates for the view being drawn. Thus, if (\,q) represents some point on the image plane, the corresponding CRT position (IX,IY) is given by IX = FIXL + (~ – ~L)(XCONV) and IY = FIYB – (q – q~)(YCONV) .

N. Chopper In engine calculations, the cells can become very thin in the z-direction in the squish region between the piston face and cylinder head, resulting in severe timestep restrictions. To alleviate this condition, subroutine CHOP is used to strip out or add planes of cells across the mesh above the piston, thus providing a control on cell heights in the squish region. When the chopper is used, grid lines are required to be vertical (z-direction) through the squish region, although radial grid lines are not required to be horizontal. An example of a mesh that meets these criteria is shown in Fig. 22. The volume-of-overlap logic in the chopper accounts only for vertical displacement of vertices. A truly arbitrary volume- of-overlap algorithm for three-dimensional space would be significantly more complex and has not been required for any application so far. The input quantity NCHOP specifies the minimum number of planes to be left in the squish region and is usually equal to 2 or 3. CHOP automatically removes planes on the compression stroke and restores them on the power stroke. The parameter LNZP prevents the number of cells added on the power stroke from exceeding the available storage. The quantity DZCHOP, calculated in RINPUT, is a function of NCHOP. The 6Z’Sin the squish region are always uniform, and each cycle tizis compared to DZCHOP to determine if a plane is to be added or deleted. The current algorithm for DZCHOP is fairly conservative: in typical engine calculations that begin at ATDC = – 90°, chopping will occur between about –45° to – 27°, at which time our specified minimum number of planes (NCHOP = 3) is reached. o. Dump and Restart Provision is made for running a problem in segments. If the input quantity TLIMD = 1.0, the code compares the job time limit to the time used and writes a restart dump on file TAPE8 when less than 90 seconds of time remain, and the run terminates. In addition, a dump is produced on TAPE8 every NCTAP8 cycles during a run. This feature is provided to minimize time lost due to system or hardware crash. Each time a dump is written TAPE8 is rewound, so that the last dump is the only one saved.

94 TOmfitinue-thqmobkron a-subsequent-run, the quantity iItlIfYl? “ininput data file is set equal to .the.dump-numbmz. ‘IMedurnp .flle is read as T.APE-7-..4fter readingthe re’ start dump, -~-A-11 reads the remainder of the input data file. This allows the user-to imodify datai~~-d~cal~tiatian ifdesire-d; if this is a~ne} e-me.mmye.taka.not..t~.i htro- duce inwmsistencies, &~~i+~-ca~ls-a-built= in-,wndom-numbe~elmrdturatxevera"i"p~aces in the spray subroutines. This generator is intended -inbe.asportableas possible for.use on other- cmn-- puters having different word lengths and attempts-h g&a&same.number seqgence regardle~~-o-f=cmqmter: ‘Gccasimmdly; the-user ma y iinci“it desira’bie to restart from a dump and”calculate identical numbers to those that would result from a single longer run with no restart in the middle. To allow for this, KIVA-11 includes the current seeds RANB an&R#@W ftn%he-ramiorn number generator in the dump data and ‘resets them aa.part.of. the.restar@rme&me..

Tine-authors wish ta give speciai “acknowledgment.tQ.~~= .McKinley-of C.ummins - Engine.Ccnnpany, who .helpedinformulating-someof-the finite differenceequaticm~ and their numerical solution procedure and who recommended many additional improvements to the computer program. We also thank many users of the preliminary versiims -of.the.code who have found “bugs~ an-dsuggested improvements. Among th-ese users-are John Rarnshaw ofyda}~o--l+ationa-lEn~neering Laboratory, Rbif R@itmd Rarnachan&a I_Ji.wti&r&-~~NG@rSF~SekLabQ=a&~ie~ ard Kerm%+Mtr~=uf Sandia -.Natiomal.Laboratories. Rina-lly-}w%hank Adrienne Rosefi -fo~--lmrgood-~nature-d- perseverance inteyping$he manuscr@..

[ APPENDIX A

DETERMINATION OF THE PGS PARAMETER

The objective of the pressure gradient scaling (PGS) method is to scale Up the magnitudes of the pressure fluctuations in far subsonic flows and thereby increase computational effkiency without changing other flow features of interest. The method is implemented by solving equations that are modified only in that the pressure gradient term in the momentum equation is multiplied by a factor l/az. The quantity a is called the PGS parameter and is constrained to be greater than or equal one. It is shown in Ref. 36 that if the solution to the modified system has pressure fluctuations that are small, in the sense that 8pa/Fis small compared to unity where p is an average pressure, then the modified system will have nearly the same solution as the unmodified system, except that the pressure fluctuations will be increased by a factor az; that is

(A-1) where Spa are the pressure fluctuations of the modified system.

Since in many problems 6p/F is approximately equal to the square of the Mach number,GOone suspects that the PGS method is in effect increasing the Mach number by a factor of a. That this is true can be shown by examination of the acoustic wave equations with the factor l/az multiplying the pressure gradient term. It is easily shown that sound speeds are reduced by a factor I/a. Computational efficiency is improved because the efficiency of many numerical methods for solving the pressure equation is improved when the sound speed Courant number cAt/Ax is lowered, 36and the PGS method lowers the Courant number by the factor l/a. The PGS method should not be used in problems where it is important to calculate acoustic waves accurately. In many subsonic problems, however, the acoustic mode is not important and the PGS method can be used to enhance computational efficiency. We now discuss the method for choosing a. The derivation of Ref. 36 shows that a cannot vary in space but can be time-dependent. It follows from the above brief discussion that to optimize computational efficiency one should take a as large as possible while still maintaining tipip small compared to unity. In K.IVA we choose a to maintain ~pa)p s0.04, unless the value of a so chosen becomes less than unity. In this case, the pressure fluctuations of the unmodified equations, which are those that pertain when a = 1, have become larger than 0.04, and we set a = 1 and thereby deactivate the method.

96 -

Irmmrwdetaii; .tise-algorithm for choosing-a is_the-foNimring..Eacli .cy& we.caicdiite.

whe-re

‘n— P–

Accordin~to Eql (A-l), in order to make the maximum relative pressure-fluctuation.on the next cycle equal to 0.04, we should take an + 1 = a*, where

@.3)

We have found, however, that taking aR+-I = a* can resu~t in severe oscillations in the computed values ofa. Tine algorithm in “&IVA-,which works weil “inpractice, is to take

Thus if a* > an we allow a to relax to its desired valu..a* witkrelaaxation thrne~r..The time ~ris taken to be the maximum of 20Atn and four times a characteristic acoustic wave transitthne aerms the ~mmputatiomd mesh; ‘mdseuLon-a-scaled average sound “sp_eed“c”/afi. Hence, .we take

(A-5)

w-here

97= and F and F are the volume-averaged pressure and mass density in the computational mesh. If the value of an + 1 from Eq. (A-4) is less than unity, we set an + 1 = 1.

AJ>PENI)[X B

TURBULENT BOUNDARY LAYER TItEATMENT

Wall functions are analytic solutions to simplified turbulence equations and are used to infer wall shear stresses and heat losses in lieu of numerical solution near walls of complete turbulence equations. Numerically, one accomplishes this by matching the computed fluid velocities and temperatures at grid points closest to walls to the wall functions, which then determine the wall shear stresses and heat losses. Numerical solution of complete turbulence equations is usually impractical because one cannot provide sufficient resolution. Although it makes computations affordable, the alternative of using wall functions can introduce large errors because in practice many of the assumptions are violated that one needs to obtain analytic solutions. In the first section of this appendix, we derive the wall functions used in K.IVA-11 and give the assumptions used in the derivation. In the second section, we tell how the wall function approach is implemented numerically in KIVA-D.

I. DERIVATION OF WALL F’LJNCTIONS

In this section we first give the assumptions that are needed to derive the wall functions and the simplified equations that result from making these assumptions. We next nondimensionalize the equations and thereby introduce a dimensionless wall heat loss <. The quantity

S& there-al-eino-sp=ay -m.mrw%; 6. the-dimensionless waH “heat 10ss.->- pt wherept is the kmimmvkcosity);- and 8. ‘Mach numbers are smail~ so that dissipation of turbulent kinetic energy is a negligible source to the internal energy. T%eabuve--lisl-leai3Sti-provisionalwall"functions that closeiy resembie those commonly used’ii~ cor@nctiim with the k-– c turbulence model.z~’ Assumptions 1--6are frequently vi~~ated .-a.t.~~d .pe~~t~-~~~-~a~~s-in .~n~er~~i~~m~&,iorI ‘enQIle:C~lcUl~tiOHS:SOIIie&3ie assumptions can be made more valid by providing .more-resoluticmnear. walls... For. exam, pie, the measure of-the flatness ofa wall is the ratio ofy, the normal distance.frmnihe-g~id- point%o the-wall; “b r; the-wtiii’~=dius-of’tiuwature: 13ytilminishing y, one lessens this ratio, and the wall looks .flatter to.the.flom.. TJe-validity-of.other= assum@ionsj for-exarnpb assumption 3, will not be improved with increased resolution. To obtain more universal wall f%mcticms,it ‘would-bedesi~~bie- to rehaxtiro-se-assumptions whose validity does not depend.on meslumiolution, -and-the analysis thatfollcnvscan serve as-a-basis-forf uture extensions of the theory. With assumptions 1-8 abo~ej the-k – c equations neara wall become

al’ ~_ . Jw = constant , @3-2) +]

(B.3)

and

(B-4)

-where

k’ (B--5) p=cpp.— , &

/- p Cp (B-6) K=—, Pr

(B-7)

In these equations y is the nomnal coordinate to the wall and u is the velocity component tangent to the wall. In the absence of chemical reactions, the species mass fractions are constant, and hence the mean molecular weight W is constant. Although it is not necessary for the analysis that follows and does not alter the results in any fundamental way, for simplicity we also assume that CPis constant. We now nondimensionalize the equations. The dimensional quantities character- izing our problem are the wall shear stress ~W,the wall heat loss Jw, the wall temperature Tw, the wall density pw, and the specific heat CP.From these a characteristic velocity, the shear speed, is defined by

Tw (B-8) U* = J —. Pw

The only quantity with dimensions of length is the distance y from the wall, and accordingly when a length scale is needed, we use y. We nondimensionalize u by u*, T by Tw, k by (U*)2, and e by (u*)3/y. Written in terms of dimensionless dependent variables (for which we use the same symbols as their dimensional counterparts), but retaining dimensional independent variable y, the equations become

k2 au Cpp—y—=l, (B-1’) &?Y

(B-2’)

2 ak au 2 a Cp k2 –P: =()> (B-3’) ~ [- Prk p;y~ 1+cp P~yt! ()—?Y Y

~2 c k2a& au 2 (B-4’) J-P ~Y— - +ccpk — –ce P—=o) Pr ?YY()1 81 P ()a 2 y2k E pzr=l

We now assme-the.dmentionle=..wall.heat.loss

~-= u- + ~ 7:-+. . 0.“ 1?”’

T= TO+ T1<+ ...,

k=ko+kl< +...,

where ui, Ti, ki, &i,and pi are functions ofy alone. Now note that dividing (B-2’) by (B-l’) yields

from which we immediately obtain

cn!o (B-n) —=0 dy. and

Consequently To = 1-and

Ti+l-= u+ + cl , where c~is a constant. Our strategy will be to solve the equations for an isothermal boundary layer for uo, ko, and co. These zero-order terms in

T=l+(UO+CJ<, (B-13) where we have used (B-9) and (B-12). Solutions for the zero-order terms are

&o = [c+k – c. )Pre]-* = l/K , P &2 Cl and

(B-14) ‘o = l/Kt’ny + const .

Thus a logarithmic velocity profile is obtained, and the analysis shows that the Karmann constant K is related to the other k —e model constants and cannot be independently specified. For the standard values of the model constants given in Table II, we have K = 0.4327, which differs slightly from the commonly accepted value of 0.40.61 Determination of the constants in the velocity formula of (B-14) requires consideration of the laminar sublayer, wherein the laminar kinematic viscosity vt becomes important. A dimensional argument gives

[B-15)

where the constant B has the experimentally determined value of 5.5 for smooth walls.Gl In dimensional terms, the wall functions fork, e, and u become

A?= c–*(U*)2, (B-16) P

Cwi .J.2 1 (U*)3 p k &=——.—— (B-17) K Y KY’

102 and

From (B: 1-6),we obtain

which -is the boundary- ~mnditionusedikr. the-k=equation4 n_KXV_AIL.Equation (B- I?) is. ‘used-diirectly todetermime the vzdu-e-ofs af the centers of computational” ceils next to -walls. 13ecauseof@-1-7), -whenever”a-length scale L-is needed in the ICilWA--~-inputor equations, it is.relatecLtdand.a hy.

We do not use (B-l%) directly ftir the velbcitywall-function beeause this would require iterative solution for the unknown shear speed u*. Instead we change the independent variable by replacing_ yu*/vt by its l/7-power law valueG3

yu* yu y~ — (B-20) —=%.-.. W.,H VP j , ~~ . w-here ctW = 0;15. We obtain

which can be easily solved for u*, once y and u are known. Equation (B-21>”is oniy vaiid”in the logarithmic region, where (yu*)/vg >>1. If yu*/ve < II then we are in the kuninar sublayer-and another formula-must be used. Although the ilow in the k.minar subiayer is not trui~ h.minar, we use the.laminar. formula The transition between (B-21) and (B-22) is made at the point Re = yu/ve where they predict the same u*. Solving

(B-23) for I?c gives RC = 114. Strictly speaking, we should also not apply (B-17) in the Iaminar sublayer, but we continue to use it for lack of a better alternative. Although we obtain the wall shear stress from (B-21), note that if assumptions 1-8 are valid we could also obtain this stress from Eq. (B-16) and the computed value of k at a grid point in the logarithmic region. Some authors2b use (B-16) to eliminate U* in the argument of the logarithm in (B-18) and thereby obtain an equation that can easily be solved for u* once u and k are known a distance y from the wall. To our knowledge, no one has tested the relative accuracies of these different equations for u*. We now turn our attention to the temperature wall function. From (B-13), in the logarithmic velocity region, the dimensional temperature equation is

JwPru* T (B-24) —=1+ :+CO , T CPGWTW () w

where co is a constant whose value must be determined from experiment. Unfortunately, good experimental evidence for co is lacking, and we determine co by matching to a laminar temperature profile in the laminar sublayer region. More precisely we assume that

T JU JwPreu* u —=1+ —y .1+ — (B-25) T KtTw CLT U* w pww

for yu/vt < Rc, where Ke and Prgare the laminar heat conduction coefilcient and Prandtl number. By equating (B-24) and (B-25) at yu/ve = I?c, where u/u* = Rc+, we obtain

(B-26)

This is only a provisional value for co that must be tested in experimental comparisons.

104 II;- IfJUMEEUC-AL .IMP”LEME NT-A-T-ION-

We now describe the numerical-implementation of the turbulent boundary layer . equa-t~onssfhst- for-tkrrmrnenturn eQ-Uati on -and-thenfm%b-:intxmmlene~bggr k- ~.aQd.-S- equations. Consider a typical cell adjacent to the.wall, .asshowninEi~ 13.1 .-V-ertices e, f, g, and-h lie on the wall, and vertices a, b, c, and d are in the fluid. To evaluate the shear stress -weinee.d-Atnknow the veloc+t y u tzmgemtti-ihe- wa-11,evaluated a distance y from the wall; and the hminar kinematic.viscmity Vt..TJfitangentiid ..velocity-u.ls evaluated bJ’-

u=ii(ua +ub+uc+ud)i.

Equation (B=27) assumes the nmrmd velocity at points a; b, c, and d is negli-gilble, so that the-tangential component may be replaced by the magnitude of the velocity. Tne distance y from the wall is.calculatedly

(B-28)

where Aa is the area vector of the face of the cell that lies on the wall. The kinematic viscosity is evaluated by

11 (g’!) “’-iiir‘ 7 ‘vP =— (B-29) . P’”

c b

d where pair is given by Eq. (24) and p and T are the density and temperature of the cell. The shear stress ~Wis evaluated using Eq. (B-8) and either Eq. (B-21), if yu/Vts I?c, or Eq. (B-22), ifyulve < Rc. With rWthus determined, the product ~@At gives the total change in fluid momentum occurring on a timestep due to wall friction associated with the cell in question. One- fourth of this change is apportioned to each of the vertices e, f, g, and h. These changes are effected by taking the change in momentum of vertex i (i = e, f, g, h) to be parallel to the velocity at the vertex next to it and in the fluid. For example the vector momentum change for vertex e is then

(B-30) a’i’eth.le = –+T “AAtu~llu~l> where lkf~’are the vertex masses. These momentum changes are added in Phase A. It should be noted that the momentum changes at vertices i = e, f, g, and h are only those changes due to the particular wall cell in question. Similar changes to the momentum of each of these vertices will result from the other wall cells to which it is common. The formulation given above assumes the wall is stationary. If the wall is in motion (e.g., a moving piston), it is necessary to transform the velocities into a coordinate frame moving with the wall before applying the equations. The new velocities must then be transformed back into the laboratory frame. The wall heat flux Jw is computed from Eqs. (B-24) and (B-26), if ydve > Rc, and from Eq. (B-25), if yu/ve < Rc. The specific heat CPis given by its value in the wall cell in question, and the temperature 2’ is given by the average of the temperatures in the wall cell and the fluid cell above it. The product J~At then gives the energy lost to the wall, which is therefore subtracted from the internal energy of the cell. It is also necessary to allow for the kinetic energy dissipated by the wall friction. This frictional dissipation is approximated by ~WuAAt, which is added to the internal energy of the wall cell in question. Implementation of the boundary conditions for the turbulence equations is straightforward. The k-equation condition, Eq. (B-19), is enforced by allowing no diffusive flux of k through the face of the wall cell that lies on the wall. The value of e in the wall cell is determined by Eq. (B-17) using the computed value of k in the wall cell and y equal to one- half the value of Eq. (B-28).

106 A-PFwNDm c.”

NLLMFXICALSOLUT.TQN.OF T.HE .E~IJA!NQIW GOVERNING-SPR-A-Y”DYNA-MICS-

Irrthis-appendix, we-give-the- finite-ditfference approximations to the ordinary differential equations governing droplet trajectories and to the integrals that give the rates of mass, momentum, and energy exchange between the gas and spray. We also describe the soiutionproceciure and the FO16SOfthe.subio.uEiWtifi: w:k125;:t-he-S~-ay-~~[~ti-lhtitmsare performed. The calculated spray source terms are added to the gas mass, momentum, and energy equations in subroutine I?CQUPL. The source term Oijhs,.-,. which gives the rate of mass addition tmthe gas per unit volume due to spray evapora-tion, is difference as follows:

./\

The summation is over all particles located in computational ceil (z,j,k). The quantities

.M:--=dP -.. u :F’p-a.w”pv.isiond .~J~+..~.~#.~~=fi&~:~~*.I am--.2 :.-.iw-1au~H“@””~~-~-~”&~~-~dl%i6i~~~ These may ai~fflerfrom iV~7anfip~”%ecause oft.he dropkt.collision andbreakup ealeula-- tions. The liquid density p~-is-asswned to beams%ant and equal for-all droplets. T-hc- finite-difference approximation for rLDAis gjven below. Tlie source term ~~~s in the internal-energy equation is difference as follows:

@;:k = _ L ~ N>d; rt”{(pr’4)3 Ie (N;”)~ – (r’$ [t (T; )

Vn,kAt .mi i, j.k) P ~J”

.- + (rqJ

IIrEq. (C-2) r’p,T’dp, and V’P are droplet radii, temperatures, and-velocities that have been parti-iiii~ -ed~d’u~ .to.dQ@”#. mlii-~-~as and~-b~ea~=~~s:The ..~.~a~.ity.v’P-~i~:~~ta-i ~~. the Wavitati6nal:accelerati5n upda-te~ ‘The ealiul-ation of the Phase A dmplet-te~nperature T~P~’-isdescribed below, and U’Pis calculated as in Sec. 111.C.The subscript (t,rn,n) denotes tlie-indiies~f.the -momentum (cellin.whiehpar$iele p is-located. The -velcmit,yvP~is a par= tially updated “particle velocity that is obtained by solving the following finite-difference approximation to the particle acceleration equation:

m? , V;—v P (C-3) = Dp (ll;nl~i- u; – Vi) . At

In this equation Dp is the particle drag function, which is defined below. The source term W~ks to the turbulent kinetic energy equation is difference in the following manner:

1 ,, w;k=– — N; ~ Izpd(r~)3(v~ – Vp) . u (c-4) z P’ v;kAt pe(ijk) where vpt is obtained from Eq. (C-3). The gas and droplet momentum equations are difference in an implicit fashion that circumvents timestep limitations due to the strong coupling of gas and droplet velocities. The finite-difference approximation to the gas momentum equation can be written

(c-5)

where E~k represents all contributions to the Lagrangian phase momentum change of vertex (i,j,k) except those due to spray momentum exchange. The finite-difference approximation to the droplet acceleration equation is implicit in the gas velocity and linearly implicit in the droplet velocity:

V;–v P = Dp(u;k + u; – v;) , (C-6) At

where the drag function DP is given by

~ p;k Ill;k + u; - v“ ~ (R, ~ Dp=-— (C-7) Dp” 8 Pd ~A P

The particle velocity VIPhas already been updated due to gravitational acceleration. The drag coeftlcient CD and droplet Reynolds number ReP, which are defined in Sec.11.B, are evaluated using time level n values of the fluid variables. When Eq. (C-6) is solved for v> and the result is substituted into Eq. (C-5), one obtains, after some manipulation,

108 ((3-8) where

(c-9) and

depend-on-explicitly -knmvmvalues-oftbe gas and droplet variables. ‘These arrays are

Cdi.Xllati”iilSUhlWthf2 .PM.~.I _~.h~n.-th~ ~~~~~ ~..&~&xJ~~@+~Jr. ~~~+~~wfi~ ~~~~~~s~ ---- droplet’:e!oeiti~ -a~~thmmmputed fron--Eq: {C-6) in-submutir~e-PACCEL. The particle radius and temperature changes. are.obt,ainetiy evaporating the par.- ticies sequentially at constant pressure. By coupling the particle evaporation rates more closely, the sequential evap_amtio~~-ea3culation allows the use of larger timesteps than.a simultaneous evaporation calculation.. The..difference approximations afthe evapcma%icm cdculaticm .ar.e.irnplicit in the droplettemp~a%urebutexplieitin the gas temperatureanti vap~r. ma=s+fkc-tien. !h=e. %h~agFr-tr%WZ=+~e-}=-a-~-eewqmrated scxqwerrtiwiii; thti--ezpti cit.: ness can produce Unphysical”changes in the computed gas temperature and vapor mass fraction when heat and .ma.ss,transfer.r~des-t.oa=single particle.ardarge.. ~h~~ez~tthis; for -each particle wecompute an evaporation tin-restep8teUba-sed on heat and-mass transfer rates, and subcycle-the.e~ apm-ation .emlculation a number of times~qua-1 to A#8teu = AZ2U. T-heevaporation calculation is performed in-subroutine EVAP.- We ilrst solve implicitly fortheupbte-ttipiet~emperature. Tine equation we approxirnate.iiobtaihe~ from Eq. (41) by eliminating R using~q.. (NY)ancieliminating_Q~ using_Eq, (42):

0.. . Pd%r’ceTd= KtirO’ – Td) Nud – L(Td) (pD)air Bd(Td) Shd , (C-II) where

(C-12)

109 and Y*l(z’d) is given by Eq. (40). The finite-difference approximation tOEq. (C-II) iS

, U+l rd – 1’; t’rz [1 + Bd (7’;+ 1)] P P P Pd 2 k)2 cer~jp) at = K:ir(?ijh – l’;+ 1, V;u ev P Bd (T:+ 1, P

– L(T; ) (pD):, V:h en[1 + BdO’:+1)1, (C-13) P P where superscript u denotes the value of a quantity after v evaporation subcycles. We initialize T~PO= Z“~Pand rpo = The quantity 11~is calculated from the formula “P”

Y;(T;+ 1) – (qtik P Bd(T:~ 1, = (C-14) P I – y;cr:+ 1, P

Cell (i, j,k) is the cell in which particle P is located, and (~I)ijk in (c-14)and ~tjkin(c-13) are intermediate values of the vapor mass fraction and gas temperature that have been modified due to evaporation of particles with subscripts less than p and evaporation of the current particle p on subcycles less than U. Formulas for these quantities are given below. The heat conductivity .Ka~rUand mass diffusivity (pD)air” are calculated using ~~h and Tdpv. The quantities V~~Uand VAT.”are given by

V~~= 2.0 + 0.6 Re@; and

(C-15) V;u = 2.0 + 0.6 Re~Pr~ ,

where the drop Reynolds number Red, Schmidt number Sc~, and I?randtl number Pr~ are calculated using rPU,Td. Uand the intermediate gas temperature ~~~. Following implici~solution of (C-13) for T~pu+ 1, the drop radius rPu+ 1is obtained from

{n[l + Bd(T: )1+ en (pD):r [1 + BJI$+l)I P P (C-16) (r “+ 1)2= mm 0.0, (rj)2 – 81eU — V;h P [ P~ 2

110 which approximate- 438) fm.therate of-drop radius change. ‘The intermediate temperatures and species densities are obtained as follows. BefGre the particle e’Japo~Q-tionealeulation -vwinitialize

n n: r .j* Wo)..k = I;k + —-. P;k

‘dlmr.as-each-.p_afi.icle is evapmatex-”we rnedif~.%he.-abo:ie=mrqs-by

and

(C-18) if particle.zq liesin-cell.(i, j,k).. Finally., .we.caladate.

(%-l).k=(”al)ijkl’l%i~k and

?.., = T:k + lJa These new intermediate temperatures and species densities are then used when calculating the radius and temperature changes for particle p on the next subcycle or for particle p+ 1 if we have completed the evaporation calculation for particle p. The choice for 6teUis based on the idea that the heat transfer to a computational particle in one timestep should not exceed some fraction of the energy available for transfer. The heat transfer rate to a computational particle on the first evaporation subcycle is approximately

QP = NuPKair (?ij& – T; ) 2m-’N’ (C-21) PP” P

The gas energy available for transfer is approximately

Eg = (Cp);k (?..k – T; ) p;kv;h . (c-22) P

The criterion for Stevis thus

(C-23) Qp~teus P g’ where f < 1.0. Using f = + and substituting from (C-21) and (C-22) gives

n (c )?. ‘;kvijk p ilk ~te” < (C-24) NupKtir 4rcr~N~

A similar criterion based on mass transfer considerations gives

(C-25)

Since ShP = NUP < Vsh, where V~his given by (C-15), and (pD,)air = K~ir/(cp)~~n = Fair, where Pair is the viscosity of air, we replace (C-24) and (C-25) with the single criterion

(C-26)

112 More precisely, we set 6teU= At/Neu, where the number of subcycles NeUis the smallest positive int.eger such that &teusatisfies (C-26).

A-PIW-NBIX- D-

PA-RTICLE RA-DiUS SE LECTK)N A-’T-INJECTK)N

Through input switch INTJDIST,the user specifies one of two size distributions- associated “with dropiets injected-into the’ computational mesh. If INJDIST = O, a mono- disperse distribution is used ”[ti(r– t-Oj]Ywith the injected-drop size r. given by inpt parameterSMR. This option canbe-used in conjunction with the breakup-model (Appendix F) to calculate atomization accordingto the method of Reitz.4 One injects drop- lei~with~e apalto.the nozzle radius for.hole nozzlesar equal to half th~nozzle~pening size for nozzles with pintles. The “atomization” of these large injected drops into smaller tiopieti--is--thec aleulatti”byby the breakup model. If.”INiHNST = 1, ax-squared dis+ributicm is-used for- the sizes ofinjeeted-draplets:-

f(r) = Le-ri; ,. r

... –..-.. where Pis the number-averaged ‘drop radius, which tor the distriliutitioflll-. to the-input:~duterzne”mnam-r.:~ by:

r =*r39.

ways to obtain a specified size distribution when injectingy.articles because one has the freedom to choose the number of drops per particle. The method we use, which is also used in the droplet. breakup c-alc-ulation, samples-mm$f%quently those portions of the size distribution where the most mass occurs. These drops will usually exchange the most. mass; momentum, and ‘energy with the gas. in the remainder oftliis appendix, we detail how-the radii ofinjeeted pa-rtiel~-a~~hosato obtain the specified drop size distribution- (.~__~,

In addition to the drop siizedistribution (D-1), we can define another distribution g(r) in sue-h-a-~v-a-ytha%g{r)dr is-the-probability. that a parti de has dro.m-.-witradiiii in=th-e=

113 range (r, r + ok). The number of drops per particle is then proportional to the ratio f(r)/g(r). Best resolution of the drop-size distribution is obtained where the values ofg(r) are largest, and it follows that to obtain the best resolution of the size distribution where the most drop mass is located, g(r) should be proportional to the mass distribution rsf(r) and the number of drops per particle should be proportional to I/r-a. From this it follows that the total droplet mass associated with each particle should be constant. This constant is determined by dividing the input total spray mass to be injected TSPMAS by the input total number of parcels to be injected TNPARC. The distribution g(r) is normalized to unit total weight by taking

t-’ - g(r) = Y4 e-r” . (D-3) 6r

According to our procedure for selecting radius values randomly with the distribution g(r), we must first find the cumulative distribution h(r) associated with g(r), and then apply the inverse of h(r) to random numbers uniformly distributed in the interval (0,1). The distribution h(r) can be seen to be

(D-4) h(r)= 1 –e-r” ‘11 +r/F + ~(r/F)2+ ~(r/F)3 I. “’-

The inversion of h(r) is performed numerically. We store values of h(r) in increments of 0.127 between r = Oand r = 127 = 4 raz. The value of lz(12 F)is taken to be unity. (This involves only a slight inaccuracy since h(12 7) is in reality greater than 0.997.) IfXX is a random number in the interval (0,1), we find that value of n for which

(D-5) h[o.12; (n – l)]sxx

Then the corresponding drop radius is

r=0.12rn=0.04r32n. (D-6)

114 Consistent wi~h the viewpoint of the stochastic particle method, drop collisions are calculated by a sampling procedure. The alternative is to try to represent the complete di~~~butienof”d~op properties tiiat”ari~e citietcdiwpcuiiiiiiims. I@ exampie,. having_cal~ ~l~la~ed,the=cO~li~im.frWu@ey be~~~en-a drop associated -witl~-pzzP~ie]e-Aand ‘all “dFop_s associated with another particle, we could proceed in two ways. In the first way, we could use-the-collision frequency ti.cdcldde.the.~~bable number ofdi-ops -inparticle.kthat lunderga collisions-with drops-in the other-particle. ‘h mpresent-the-dsttibution ofcoHi- sion behavior, this number of drops would be subtract.edfrom=w&icle._A, and-one or.more new particles would be created having the properties of the drops resulting from the collisions. We tried such a-procedure with the result that we quickly -hadmore particles than mndd ‘be acmmmociateci ‘by-camp uter a$or~. k.these,cond~-

‘I’he subscripts i-and 2 refer to the properties of the collectors and droplets, N# is the number of”droplets in particle 2, and V~~n is the volume efthe c~ll in--which-both particles

Iili are located. The probability Pn that a collector undergoes n collisions with droplets follows a Poisson distribution,

— n Pn= e-z!-, (E-2) with mean value fi = VAtwhere At is the computational timestep. Thus, the probability of no collisions is PCI= e– fi. A random number XX is chosen in the interval (0,1). If XX < Po, then no collisions are calculated between the drops in particles 1 and 2. If XX —> Po, we chose a second random number YY, O < YY <1, that determines the outcome of the collision. ~(rl + rz) is the collision impact parameter b. If b < bc~, where bcr is the critical impact parameter below which coalescence occurs, then the result of every collision is coalescence. If b z bcr, then each collision is a grazing collision. The value of bcr depends on the drop radii, the relative velocity between the drops, and the liquid surface tension coefilcient. The expression we use for bc, can be found in Sec. 11.B. Suppose the outcome of the collision is coalescence. Then the number of coalescence n for each collector is determined by finding the value for n for which

n—1 xPksxx<~Pk. (E-3) k=O k=O

For each collector drop, n droplets are subtracted from their associated parcel, and the size, velocity, and temperatures of the collector drops are appropriately modified. If there is an insufilcient number of droplets to have n coalesce with each collector, then n is recomputed so that all Nzn droplets coalesce, and the particle associated with the droplets is removed from the calculation. There is a timestep limitation associated with the above calculation of drop coalescence, and this is that the computational timestep At be small compared to the collision time At~ for the droplets. The latter is given by

1 N; (E-4) —=vd=— n (r: + r~)21v1 – V21 , Atd V;k

where Nln is the number of collector drops. When At << Ltd, the probable number of droplets that coalesce in a timestep, which is VdAt N2‘, will be less than N#. Hence, most of the time the number of droplets in particle 2 will not be depleted in one timestep due to collisions.

116 This timestep limitation is much less severe than that-we would have required 11we had’%.l.limrefii~.-one coales~<.nce per tiines%ep. With one ~malescence-per timestep, we would have had to compute v6t << L wherev is g&zrL by Eq. .(GQ,. Theeqm+-ims fore;. aml:v~ diIffer in that: the number ofthopN4WV& is used in mmputhl-g-v. In Appendices D andlrit is shown thatwe usua-lly have -N# >-> -.l’-+ln-amHnencev >> v~. Suppose now that the outcome of each collision iwgrazing rwdlision. Inthist-as~ only-orm+ollision .is ealeulatecHbrwzchdmp: Tlnis-in$rodlxxisan addi~iimal-t%-msteq ~anstraint-that~ tb~=mall ~mmpared-’t thecollision-times between drops of neariy equal- ,— . . . .. Sizej wnce grazin.gcolhs~ons usually ocmu.rh~@~~Y._d~.~E~~.~~~r]~l~yaa.~.~~~.. .C..=.=.bu Laaklcg collisions are calculated between N pairs of drops, where N is the minimum of N1 n and ~12~Fn. ml~m#J collectors and’diwpletxrare-then returned”to their particles in such a way that mass, moment- and ‘emr.~..8_re.eQQse.m7ed,

In this appendix we-describe-the numerical -solution.~o~~ure foethe eqrmtions=g-ov: erning spray dfipi~osci-litition and breakup. A detailed-description of the.br-kup-nmdel pan be found in Ref. 35. ‘h ealetitiedraplet. oscillation.and %HWakUpiwe -require two addi= tional =particlemm ys-yz~ami jP_ Tine quantity.yp iiqprqmrt.iimal to -the dispkwrnentof-the droplet’s surface from its equilibrium position, divided by the droplet radius. Droplet breakup occurs if and only if yP exceeds unity. The time rate of change of yP is yp, and the time rate of change of jP_is given by Eq. (45). To -uplate-the-vaiues ofyP and y~ each computational-cycle, we make use of the exact- soltition ofEq, (4-5).assumi-ng~wnstallt-co-efflcien@:

(F-1)

‘where

(F-2)

117 is the Weber number, u is the relative velocity between the gas and droplet, a is the surface tension coeffkient,

2 edr2 (F-3) ~d=– _ 5 P~ is the viscous damping time, pt is the liquid viscosity, and

a 1 (F-4) CI.)2=8— ——~2 Pdr3 ~ is the square of the oscillation frequency. For each particle we first calculate We, td,and ~2. A value of ~2 —< () occurs only for Very small drops for which distortions and oscillations are negligible. Thus if @ ~ O,we set yP~+ 1 = yP~+ 1 = O. If GJ2>0, we next calculate the amplitude A of the undamped oscillation:

(F-5)

If We/12 + A —<1.0, then according to Eq. (F-l), the value of y will never exceed unity and breakup will not occur. Most particles will pass the test We/12 + A ~ 1.0, and for these we simply update Ypand yp using Eq. (F-1):

fl+l _ w’ (F-6) Yp - ~+e.r,GAti,~)[(yj- :)cosoA,+ ~(,J+Y~j,%)si.oA,) and

We t-z .n+l = We (F-7) Yp ‘+1 /td + exp(-At/td) j; + ~ MJsaAt-co(y~- ~)sintiAl\ ) [( ) ( G–yp d

If We/12 + A >1.0, then breakup is possible on the current timestep. We calculate the breakup time t~u assuming that the drop oscillation is undamped for its first period. This will be true for all except very small drops. The breakup time t~Uis the smallest root greater than tnof the equation

118 :+ Aces Ico(H”)+Q] .=1 , (F-8) where

and

If.time in+ ‘-is-less-than tbu, then no breakup oc~urs-om-the-cwent”.iimestep, and “we-use Eqs..(.F-@ ZmliL(l?-’z)tA=u#a!% jy,aadjp Breakup is calculatedonlyif t~U—< P+ ~. When this is true, the Sauter mean radius raz of the product drops is calculated from Eq. (31), and Eq. (32) is used to calculate the velocity w of the product drops norrnai-ti the relative velocity between the parentidrnp and gas. When evaluating r3z and w, Ypis evaluated at tbu using_Eq, (F-1). The radius r~~~. of-th~praduet:drops is the~~-ekosen-ramdomly from ax-square di~tributiimwith Sauter meamradius-r~~. h .ehcm4B&n~M-wesa.rnpltimokf’r%qmmtiy. from %hose-portions of’tlie x-sqgared disiritmtion where +Aemosimass nxides~ aS is deseribed- in .Amem%x---- 9.- 3% ~wnserve mass; the--numlxxofdrops N associated “with the computational particle is adj.ustedaccordin~tc.

We also add to the particle velocity a component.wii,hma~it.ude w anddires%ienra~de~m, ]Y-chosen-in-a-~]an-e-normal to tfie relative velocity vector between the p_arentdrop and gas.. !This.prncedure does not conserve .mmnentumin detail; but itdoesw-onthe average. Following breakup, we assume the product drops are not distorted or oscillating, and. ,.— a-cmralnglj we-set:yp%+~- = jP:-+ ~-= O.

119 APPENDIX G

CALCULATION OF’ DROPLET TURBULENT DISPERSION

Turbulence effects on the spray particles are modeled by adding to the gas velocity u a fluctuating velocity U’p, where each component of U’p is randomly chosen from a Gaussian distribution with standard deviation _ and k is the specific turbulent kinetic energy of the gas in the computational cell in which particle p is located. The fluctuating velocity U’Pis a piecewise constant function of time, changing discontinuously after passage of turbulence correlation time ttUr~,which is determined by Eq. (37). The sum u + U’Pis then the gas velocity that the particle “sees” when calculating its drag, heat and mass exchange with the gas and its oscillation and breakup. We also subtract from the turbulent kinetic energy the work done by the fluctuating components in dispersing the spray droplets. For each computational particle, we chose one of two numerical procedures for solving the equations of this model. The choice depends on the relative magnitudes of At and tturb.When At < t~urb,which is most often the case, it is possible to solve for particle positions XPBand velocities Vp~ by straightforward difference approximations:

x:-x; (G-1) = Vn At P and

v: -v; (G-2) = DP(L$ +U; –v;)+g, At where Dp is the particle drag function (see Appendix C) and the particle is located in momentum cell (i,j,k). The gas turbulent velocity U’p is held fixed for a number of computational cycles k such that

tn+k-l – tn (G-3) where U’p was last chosen on cycle n and tturb is the turbulence time evaluated at the position of the particle on cycle n+ k. Section IH.C gives the method used for randomly sampling for U’p. The velocity U’p also enters in a straightforward fashion in the difference approximations to the equations for mass and energy exchange given in Appendix C and the difference equations for droplet oscillation and breakup of Appendix F.

120 ‘When A-t”> tiu,~it.is.no.lon~r. ~titie.ta-la=.the= difference approxirnatiwx- because the particle “sees” more than one turbulent velocity U’P-—on the current cycle. ~ossitie-=a~-aches~~=this~robiem- are-‘R restri’c%A’ti”ti-~be-some fracti-on Of”~~ur~ or to ~cycle.the.ti~~e~ations.lming subeych.timestep ~tsuc-h that M< t~ar~.‘I%ess methods are computation ally inefficient, however, when tlurhis smaller than At, and are unusatile when tturhbecomes Ord@S Ohagnitude Smailer than Ant. Char..apprmachto this -problem fomak=s-sem~aevaraey to obtain mrnptitationaleffi- ~ie~~_y._. W-=.> Q~Y~, “~we.~Q~~c .r~nd~. v~l a~i@ and ps~l~iea.:ckaP.ge#’r9mA=pzb ah$]ity- clistributiims-that we have dixi~edss””far the dioplet turbulent ve-locity an-dposition c-hanges~.T=hu~..ti+*&.ti-b.w=’~M.”is=eelatik’e”*A:t~r~; cmrmethu&rq@%SiR5

Choim &0K4JT-!W4M%HX141ZB>WWdXMH?.Wll-&MH*, i3WAA&f3%Ri~i~- %~s–*fi-*ti-leYfi-ve-’mci& change and one to determine its turbulent position change. This approach is-inaccurate-in several respects. First, when-At > ttubi .we ignore the effects of the fluctuating velocity u‘P on heat and mass exchange and on droplet breakup. Second, in deriving the probability distributions for turbuient velocity and position Gliamgq -weliave =am--dmekthat~bhe-b~”um~tion D;J k, .amd=tjurh are constant for a given particle on the current timestep. In particular, the effects of a nonlinear drag law are not ineludedin th-~probability distributions. T-herea~pears t-obe no “alteWmdh-b-th~- assurn@imr of-a-linear cirag..iaw bec-ause-use afa.nmdbar “M. rend~rtifitr.adahldlie. probi&m ofd&riviti-g~obaMlity distributiomfor turbulent velocity and position changes. T-hederiva$icm ofthedis%ributiof is~@venin&~: S3; .and:hem--we-only-~umaulm- the-resuits: Tne- assumption ofa hear drag law aHows us to treat.each.componemt~fthe velocity and position changes independently. For each component the distributions are Gaussian; the velocity distribution has variance

an-d:the p_osition distribution has variance

~ Zt (3, 0:, = t~urbAt – +[1 –exp(-D9At)l+ ~ U2 , (G-5) [ .— q62 } ‘P

where 02-= ~lz. An additional quantity, a turbulence persistence time trmr,is used in ehoosingthe-tuAulentpos ition-ch1~e--antentem'ue=use-tb-e distrillution ofvelocity and. position changes are not independent. Thisquantity isgiYemby- (3:, L~U,Jl– eXP(-DPAOl– — z) (T2 P“ (G-6) t= per 0:,/02

When At > t~Ur6the particle velocity and position are updated using

-x; .; 6X’ (G-7) =v; +— At At and

“B _ Vn PP (G-8) =Dp(u:k–v:)+ g+:, At where 8x’ and tiu’ are the turbulent position and velocity changes. First, each component of 6u’ is chosen from a Gaussian distribution with variance OU’2. Then 6x’ is calculated from

8X’ = twrm + 6X; , (G-9) where each component of bx’h is chosen from a Gaussian distribution with variance OX’2 — tper%u’z.

APPENDIX H

THE VARIABLE IMPLICITNESS PARAMETERS

Variable implicitness is used in KIVA-11 when differencing the diffusion terms and terms associated with pressure wave propagation. The amount of implicitness is chosen, in part, to ensure numerical stability of the difference approximations to the individual terms in question. If stability were our only concern, fully-implicit schemes would be used. Computational efficiency can be gained, however, by minimizing the amount of implicitness. When the timesteps are small enough, KIVA-11 will automatically use stable explicit schemes for which no costly iterative solution is required. When implicitness is required, K-TVA-II uses a partially implicit difference scheme in whwh there is some

122 weighting of both the old- and new-time values of the solution variable. It has been our experience that most iterative procedures for implicit equations, including the conjugate resJdtial:method ”d*d-in .RIIVZA-II;~umvergemore slowly for fully irnplieitthan for partially implicit schemes. A fiuily implicit scheme is only used”by KNA-11 in the limit. of” an “Mh-ti&elj l&rfge_tiiBWzp; .Iil -w-~”[email protected]:’&:mdym~ :-wHcktisfcLm.K&: the implicitness parameters are 13aseck Imaclditiorrto rnotivatingtk-forrns ‘we-usc-f%w+ke- implicitness parameters> the analyses are.interestingin that.they reved.the.natnm.of. some of the numerical errors inherent in KIVA-11 solutions. ‘~e-fimt-&e-the-anatysis-fOr determiningthe implicitness parameter @~for the pressure gradient terms. The form [email protected]%y considering tlie.KIXA,lI finite difference equations applied to the problem of one-dimensional inviscid acoustic wave motion in a gas with nearly uniform density p. and pressure po. We use a computational mesli with uniform cell ‘si~e & and ‘cross-secti~nal ‘area A. IE cme diineriskm th”eapproxi~ mations to the vertex momentum equation (78) and cell face velocity equation (86) both reduce to the same form:

In (H-1) uj+ ~is the velocity at a vertex or a cell face between cells~ and~ + 1. All Phase B quantities equal their advanced-time values (e.g., PjB = Pjn + 1) since convection terms are negligible for acoustic waves. An equation for tiie.pressure-ii-ohtti-nefiy combi-ni-ng Eq. {102) for-the celhdurne change and the linearized form (112) of the equation of state:

(H-2b)

In (H-2) VjB is the Phase B cell volume. and VOik the initial-uniform.cell .vcdurne..~o. = A Ax. Combining (H-2a) and (H-2b) gives

n+l n.+ L a.+i_ p , –-p; ai4A—ui_4_ Jr ‘+yp ‘“= “=0. At o Ax. By using (H-1) to eliminate the velocities in (H-3), we could derive and analyze the difference approximation to the pressure equation. Our approach is to keep (H-1) and (H-3) in their present forms and solve for both u-+ ~and Pj. Numerical solutions of (H-1) and (H-3) can be found in terms of the Fourier components of Pj and uj + ~. Substituting the values

P: = An YPO~P (ikj Ax) and

n (H-4) ‘j++ =Bncoezp[ik (j+~)Ax] into (H-1) and (H-3) and solving for An+ 1 and Bn+ 1yields

1 + a2(l – @p) ~ n+l A An 1 –a2f$p 1 –CZ2+P — , (H-5) n+l a 1 ()(B )() Bn 1 –d~p 1 –a2@P \ wlfere

a=–2i C!~sinqs12,

coAt c*=— Ax’

1 YPO co=— f —2 a P* and

y=k Ax.

Numerical solutions will be stable if and only if both eigenvalues of the above matrix are less than or equal unity. The eigenvalues L of the matrix in (H-5) can be seen to be

1 – 2(1 – @p) c~sin2 qs12 + 2c~ sinq5/2 J(l – @P)2C2 sin2 qd2 – 1 s A= (H-6) 1 + 4@p C~sin2y12 124 If (1 – @P) Cs <1, the two A are complex conjugates and have common magnitude:

{~.7) i hi =-H -i 4Q”PC?:siTL2@21- ‘- .

Thus $hese.hwne isstable if.(1 —+P)C. <-l:or, equivakmtly, if@p-.—>-1 – UC.S. ltiotivated-bythis-result; we calculate the value of $~ based “onthe itwiiCourant number C’s:

(H-8)

where

Ax = min (Axi , Axj , Axk) ,

Ax::=*[lx;-– x:~~+ lx;. – x;i21-,

(H-it)) and the xi~’s refer to the vertex locations of cell (i,j,k) as numbered in Fig. 2. The quantity ~is an empirically determined safety factor. The above analysis gives f> 1.0 as a suffi- eied-cwltition-for~tatiilityinone-dimensional “pmWlems-with-unifom-ceilti~sa& material-properties. Using f = 2.5 has been found to give stable results in all our test @]&c&*;erls. When C. < 0.4,.@P will-he zero and an explicit difference approx-imation will be used by KIIJA-H. Thec~llfac*=velocities-am-fimt+ound=fl-om WI. (86); and then the Fha-se B-ceil volumes and Phase B pressures are found from Eqs. (102) and (112). Since the magnitudes of the eigenvalues give the amount of numerical damping, Eq. (H-7) shows acoustic waves are not numerically. &rnpe&by thik.esc.plid~kiherne, .aitk-mglhthere. is some numer- . icai di sp&wsion-ot+wous+ie kvaves. ‘TW3 R the difllirence scheme used:iil-tlie-oti~inai ‘KlIA- codel to calculate pressure.~ve.~-pagatim.

125- We now motivate the choice for @D, the variable implicitness parameter for the diffusion equations. The form of @Dis indicated by an analysis of the KIVA-11 difference approximation to the one-dimensional diffusion equation with constant diffusion coefficient v:

Y;+l – Y; Yn 2Y; + Y;_l Y;:; – 2Y; +1 + Y;:; J+l — (H-n) =v(l–(#)D) + +D At [ Axz AX2 1

Substituting

Y; = An exp (ikj ~) into (H-n) gives the following equation for An+ 1“.

1 + 2(1 –@D)cd(a-JsqJ– 1) (H-W _. . A ‘~l/An = 1 –2@Jcosy-1) ‘ where

VAt cd=— AX2 and

y=kAx.

Now we require that the approximation not only be stable, but that the amplitudes not change sign:

n-l-1 A (H-13) ()<— =1. An

It is easily seen that the right-hand inequality in (H-3) is always satisfied. To satisfy the left-hand inequality, since the denominator of (H-12) is always positive, we need the numerator to be positive. The numerator is minimized for Y = rzand has the value 1 – 4(1 – @D)C~. This leads to the criterion or

1 QD=-I- ~ . (H= 1-4) ~.

Motivated by this result, we compute $D from the equation

where

and Axi, Axi, and kk are defined in (H-1 O). In (H-17), As.is the ratio of the second to first cQefTIZ-ititsQfvi&QQiy; 32 ii+:theStimidt number; .and-lr., -Pi-k,and-.Prt ar.e,l%andtl num, be.rs fordiffusion-ofheatY turbulence kinetic energy, and-turbulence dissigg-dkn~--mte: For real gases; the arguments in- (II-17) are normally near unity, and one value-of @D is-used to calculate diffusion of all quantities. If one has an application for which there is a large variation in the values of’these arguments, then it is advisable to use separate vaiues of”- *D for diffusion of each quantity. Note-tktif we-had requiretijust stability of the difference ‘approximation: .thatis. ~0 ?Ak+ .4P + ~j~? —>. _ ~j from .(H-1~.)it w~~ld ha-v~bw~ sufflei~t

or even OD.= *._ ~isi"atter-Mheme-igt"ne-i5~aK.IN"icQl"~me.thQd?i" anu:iS wwmd-cmder. aec’uratein time; Although more am.mate in time; the &ank-~Nicokon method and-other methods that violate (“H~14-)have “overshoots” “and“are inaccurate for large val”ues of”-C;. APPENDIX I

KINETIC CHEMICAL REACTIONS

Here we specify the procedure for evaluating the progress rates 6A for the kinetic chemical reactions. Since there is no direct chemical coupling between different cells, we may focus attention on a representative cell (z,j,k) and suppress the subscript ~k. The progress rates chrfor the kinetic reactions are computed under the assumption

that, for each reaction r, every participating species is either inert (amr = bmr) or appears on only one side of the reaction (amrbmr = O). We first calculate the quantities

‘f, = $ ~[ (;m/Wm)amr m

and

%r = k~r n @JWm)bmr, (I-1) m

where kfrn and kbrn are evaluated from Eq. (17) with T replaced by Tfl and ~nl denotes an

intermediate value of pm that has been updated due to kinetic reactions < r but not reac-

tions >r. We next identify the species, call it species K, for which

Wm(bn,r – ant, ) (i2fr – QJ /p m

is a minimum. This species is called the reference species for reaction r; it is the species whose densitY is in greatest danger of being driven negative. Once K is identified, we define

AB = FK + ‘tw~[aK~ !2fr + bKrQbr] (I-2)

Then tirA is given by

FK(ATIAB – 1) if = AtWK(bK, – aKr)

This prescription makes the part of Jp}J~tthat is due to reaction r linearly implicit in pK, I which prevents pK from being driven negative no matter how large At k. I 128 Irrthis-appendix-we describe the two ‘procedures for evaluati ngtkie.pmgressrates. cbr-4fix the equilibrium reactions. For hydrocarbon combustion, a fast, aigebraic solver is provided in subroutine CHIViQ@Ij~or more general-circumstances, an iterative.solyer. is. provided in sulmo.utine HUiMZQ, The dkmieeof subro~tin~-is-d~~tiine~by”inpatiqag

KWIKEQ: If-KWIKEQ -=0; CHEMEQ is-used; if KWHSEQ- = i; CHMQGM-is used. in the latter case one does not need to input the stoichiometric coefficients and constants used-’t compute equilibrium-cmtiarltsin-Eq, (19); these are stored in data statements for t.he~pe~ifi~.W.t..~’_tiQaGtiGnS=l~edilti..CHMQ@&LW%!!!&t describ+ ~hemer+gene%al, iterative solver.

I* SUBROUTINE CHEME-Q

Hemwe-&scfib-the-procedurein subroutine CIHKMiiK-for evaluating the progress rates cbrAfor the equilibrium reactions during Phase A. The cbrAare implicitly determined by the reqtiirement that the Phase A species densities pm~ must satisfy an approximation tath~e~tilibrium .constraint conditions of.lTq. (18):

where

and-T and-TA -are partially -u@dtettemperatures-t hat- will be defined shortly. In (J: i) we have fineariied”f% KCr”aboutits starting value $; (The spatial indite ijk will be suppressed tlirougficmt .tliis appemlik~ as there G-m direct ehemieal muplingbetween different ~~lls-.) These constraint conditions constitute a coupled nonlinear equation system which is solved by afi iteraQve-procedfifie.? s--T;heiterafionschaeused -is-azr hrrprovcmren-tover the eariier sehwne=used -in=CON~CI~A&-WRAxiZ-]Qarz&includ~s%he-effectson the e-quiliiiri urn constraints KcF-”of”heatreiease.from the eq.uilibriurnreactions. ‘iZ For. . simplicity-, we assume that the fuel species.(speciesl), .of.which.thesp-ay-particles are ~mmposed, does not parti - eip ate hnany G1-tke:eal.ti”li”briiml~rfmctifis

129 The iteration scheme consists, in essence, of the following ingredients: (a) preconditioning of the equilibrium constraint conditions to make them more nearly linear in the progress variables, (b) application of a one-step SOR-Newton iterationG3 to the preconditioned system, followed by (c) switching to a full Newton-Raphson iteration if the simpler SOR-Newton iteration fails to converge in a specified number of steps. If the equilibrium reactions are weakly coupled, convergence usually occurs before the full Newton-Raphson iteration is called into play. However, if the reactions are not weakly coupled, the interaction between them is properly accounted for by the matrix inversion in the Newton- Raphson procedure. Within this appendix, we denote by pm and ~ partially updated species densities and fluid temperatures that contain the contributions due to kinetic chemical reactions on the current timestep. ? is given by

where the sum is over kinetic reactions.. These values serve as initial values for the iteration procedure. The final converged values of the species densities are the pmA. The species densities are related to their initial values by

ant,)0., (J-2) em = pm+wm~(b rns – s and the partially updated temperatures are related to there initial values by

(J-3)

where OS = At hs and the summations are over all equilibrium reactions. We denote by USAthe values of us for which pm = pmA and T = TA. The values of h~A are then simply ~8A = m~A/At. Because of Eqs. (J-2) and (J-3), we may regard Eqs. (J-1) as a coupled nonlinear system of equations for the unknown quantities @S‘. Since these equations will be solved iteratively, we introduce an iteration index v, which will be displayed as a superscript. Thus the approximation to USA after iteration v is denoted by QSV,and the corre- spending approximations to pJnAand TA are

130 and

T’ =?+f~ (@j(pnC;) (J-5) I s I

It is understood-that co.s0“= 0-, so thatpk o”= pi and-’~ = ‘fi It will-also be necessary to refer ‘Q intermediate species densities defined by

andintermediate temperatures ‘T(v,s) ddi”ned -by

‘Whf3R”iY is the numbenfequilibr+um- reactions. One further notational convention will prove useful. We introduce a vector p = (pl, Pj> ...9P-MS,T) -who~e ~~mp~nents am the SPeGieSdensities P.h and-temperature T. F~~~= tions of the pm and “’Tcan then be compactly written sirnpiy as functions of”the vector vari~ able p. The notation pVrefers; ofccmrse, to the vector-whose components are the pmVand- ~~, andp(v,sjrefers to the vector with components pm(v,s) andT(~,sj. We now proceed to consider how the equilibrium constraints of Eq. (J-l) might be preconditioned”to make them more nearly linear in the m~. The first step is to identify the principal or-dominant dependence of the left member-of Eq. .(J-1) upon the m~~i.The form uflhis-quantity suggRststhat-we-&temine; for-earh-reattimrs; the-species-m for which: the factor (pfi/Wm)b~~ ‘a~~ depends mostsensitively, in some appropriate sense, on the QS. Let-this be-the-species wit~l-index m-=- p(s) and-denote- bu(s),s — aufs),sby q~, The spe&s .m.= p(~j Will ~bem!=ErredtQas the r-&Sr-ume-speeiEs fiir-reac.ti6n -s. ‘The .cbminant. dependence of the left member of Eq. (J-1) on the tiSAis now regarded as being contained ifi-the-fh-ctor-@AYi@.“ /[email protected].‘ Sifice-.~iSj it=if.i~ifiem.iti.the.(~s, .thiifiminant.cQencL en-ce-c=n-b-e-m-de- tu m-anife-stitself Iin-e-arly-by-raisin-g-b-oth-s<-de-sof Eq. ($1) to-the-p-o-wer p~.= llg~, We therefore replace the constraint conditions of Eq, (J-1)M the precondi- titmeii:coms%raitit-cmditions- It is convenient to introduce quantities F~(p) and G~(p) defined by

–b Gg =@? ~(pmiwm)”ms ‘s (J-9) m and

I?. = G-P’ – exp [p~Ds(TA – fi] , (J-1O) 8 in terms of which Eq. (J-8) becomes simply

F$pA) = 0. (J-11)

We have yet to specify how the reference species are to be determined. For simplicity, we define the reference species for reactions as the species for which the factor (pm/Wnl)h~S-a~S depends most sensitively on co. alone, without regard for the other progress variables. That is, p(s) is the value of m for which the quantity

Pm arns-bms d Pm bms-ams R— (J-12) ms ‘w— ()

R ~s = (Wmlpm) (bms – an,s)2 . (J-13)

This depends on pm, so it is necessary to specify which species densities are to be used in the evaluation of the reference species. This will be done below. In the subsequent development, we shall require the partial derivatives dFJdut, which are also easily evaluated from Eqs. (J-2) and (J-3). The result is

aF s -P, — = psGs ‘Ast – (p$DsQ/pcP) eXp @8 Ds(Tk – ?)} , (J-14) a~l where the matrix ASl(p) is defined by

(J-15)

132

— %&vitiaec&Eqs-. (J-2) an*(LT-3’)”,the quantities-F~(fi), G;(p), ad~~~(~”may.altern ati~]el~r- be regardedm-ftictibnsd.the progress variables -s j and this-will bedone without”special comment when it is appropriate or convenient to do so. The- iteration sch~e.as a-whole is structured M.f611b3v&.TAefirst.iV-0iterations-are performed }vitha-on~~ep SOR-h~ewtonalgofithn; If-convergence has notalready occurred, all subsequentiteratimm are-perfmmed-with a Newton-Raphson algorithmie~cept..as noted in the description below). We currently take NO = 7. The iteration scheme is con- s%iinwdtulitive converged” when ~GsPWs~< s fdr-ail”s. Gm-rezWy z istake~,-te-be G:3z: ‘wTe now proceed to a detailed description of the SOR-Newton and Newton-Raphson algo- I%tkms-thdt-am--used-: .Aone-s%p SOR-N%vton iteration procedure; applie&to--the-system of Eq. (J-i-l-j, taJX&heUfQrmW

where Q is the overrelaxation parameter. Equations (T-14), (J-IQ), .(J-6), and (J-?) allow us to rewrite Eq= (J-16) in the more useful form

where G~ and A$~are evaluated at p(v+ 1,s) and T is evaluated at Z’(Y+ 1,s). It.is not neces, sa~fxxa-ctually evaluatep(if + 1;s) by means-of Eq. (V~6],because ifthe pm are ccmtinua-lly updated as running sums then P(V+ 1,s) is simply the “current” value of p just prior to the evaluation.of @sV+ 1; Strictly speaking, since ps is considered constant in evaluating dF~/~0~, p. (and .—.-.. therefbre.qs) should be held-constant and not allowed to vary with V. This mi@t seem all the-i~ advisiideimview -oftfie-fact-thatp~ varies discontinuous~y with the p= or ~~. In practice; ho.wewe~5.-we:h-avefound that convergence is-slightly accekrated-if ~~ and q$in Eq. (J-1.7) are allowed to varyby reevaluating p(s) in terms cfthe P;m(v.+... us.) or.every, iteration No-problems have ydleeme~erienced.”ifi. dbitig soj but if such pr3Mems were. to occur it woui”d”merel”ybe necessary to ho~dp~ and q~ fixed with the values determined by the initial .spw~-iadmsities-~w: Tine value of co~%’+1 – ms”~given by Eq. (J~lT) is subjected to the restriction where

~in Wm(a – -1 ms bm~) 66)rnin _– (J-19) s I m’ pn,(’v+ 1,s) ‘

–1 ~ax Wm (a – bn,s) ms tkll- = (J-20) s [ m Pm(v + 1 ,s) are the minimum and maximum values of QSv+l _ COSVthat preserve the nonnegativity of the pm. A standard Newton-Raphson iteration applied to the systemofEq.(J-11) yields

~ [aF,(m;,.... ti~)lao~l(O;+l - 0;) = -F,(GI;,....mj). (J-21) t

Using Eqs. (J-14), (J-1 O), (J-4), and (J-5), we may rewrite this as

~ [A (J-22: St– (D,Q/p CP)I$ q) @$Ds(?’v– ?)}] ((J);+1– (i-))= (/s[G: (MJI Q)$DS(TV– m}– 11, t where GS and AS~are evaluated using p~. To obtain the OS”+ 1, it is necessary to invert an N X N matrix. This maybe done using any of the standard methods, one or more of which are usually available as modular library subroutines in large computer centers.

Again, ps and qs should strictly be held constant in Eq. (J-22). In practice, however, we find it slightly advantageous to allow them to vary with v by reevaluating the reference species in terms of the pmVon every iteration. In spite of the preconditioning, Eq. (J-22) occasionally yields values of us’ + 1 that drive one or more of the pm”+ 1 negative. (In particular, this may happen when a single trace species of very small concentration is involved in two or more reactions, as the matrix then becomes ill-conditioned.) When this occurs, the values of OS”+ 1 – ~SVgiven by Eq. (J-22) are all reduced in magnitude by a factor a(O < a < 1), and the pm”+ 1 are recomputed accordingly. If any of them are still negative, the reduced values of coSV+ 1 – USVare further reduced by another factor of a, and the pmV+ I are recomputed again. If necessary, this procedure is repeated up to Na times, whereupon if negative values of the pm’+ 1still persist, the Osv+ 1 given by Eq. (J-22) are simply discarded, The iteration is then repeated as an SOR-Newton iteration, with the QS‘+ 1obtained from Eq. (J-17). We currently use the values a = 0.3 and Na = 6.

134 II. SUBROUTINE CHMQGM

Subroutine CHM~GM-utilizes an algorithm devised “byNieintjes and Morganz~ for the solution for the simultaneous equilibria of”sixxe~tibntirn~l.an t.in-hydro~arbon oxidation:

N2$2N

2H20 + 2H2 -t 02

‘m-mntmi~t=taCXii!3M33Qj;which so”ives for reaction progre.

(J-24)

where the sum is over all kinetic chemicai “reactions. Because the formulation does not include the effects on the equilibrium constants of heat release from the equilibrium reac- ticm~ srrall.cycle to Cycle osei lla-tions=lnkmperature a-ml.sp~-i~--cmnc~nt~-atio~m-can--oecul- in some-aTpiications. ‘z These oscillations are small in most hydrocarbon combustion problems, however, and it is better to use subroutine CHMQGM-instead of CHEMEQ because itis-muehfaskr. The”increase in speed of subroutine LCHE?IIG2-iSachieved-because the ten equations for the concentrations.are. algebraically reduced-to a much simpler systemj which-can be quickly solved. The details of this reduction can be found in Ref. 24, and here we only summarize the simple system that is solved. First the equilibrium and element- rmnservation equations involving-nitrogen; which are uneotipled -fromthe rernaindernf- the system, are easily solved for the concentrations of molecular and atomic nitrogen. The remaining eight equations are then combined algebraically to obtain two cubic equations in two unknowns. These are scaled concentrations of atomic hydrogen and carbon mon- oxide. The simultaneous cubic equations are solved by Newton-llaphson iteration, using the scaled concentrations from the previous cycle as a first guess. Because of the manner in which the equations are scaled, it is always necessary to have at least trace amounts of carbon present. The values of reaction rates d,A are not needed when subroutine CHMQGM is used: The mass and energy source terms are difference by

(J-25) and

where pm is the initial density of species m before equilibrium reactions and Eq. (J-26) follows from combining Eqs. (20), (21), and (22).

APPENDIX K

CALCU ,ATION OF VELOCITY GRADIENTS AND VISCOUS STRESSES

According to Eq. (4), evaluation of the viscous stress tensor requires evaluation of the velocity gradients. These gradients and stresses are taken to be cell-centered quantities and are considered uniform within regular cells. Velocity gradients are obtained in the following manner. We begin with the identity

au~ — =V. (utem) , (K-1) ax m

where Ueis the velocity component in the tth Cartesian coordinate direction and em is the unit vector in the mth Cartesian coordinate direction. After integrating Eq. (K-1) over computational cell (i,j,k) and using Gauss’ Divergence Theorem, one obtains

136 where S~k is the surface of cell (z,j,k). The left-hand side of this equation is difference as

and the right-hand side is approximated by

~ (u,)aem Aa , (K-4) a

-where:(ut)a is-.the.equal-wtighted average of the fciurvaities Oftit -associated-with the four vcr~ices bordering face a, and Aa is the outward area projection vector of face a. .After.eai~-uiatitig-th%e~~doeity gradients, the stress Lmmponemtsin-cell (i, ,~,k)are obtained from

(K-5)

-where-thesuperscript-v denot~-.to.tim~-’~vdtitv.eiociiyiy. fieid;

~n tlhis appendix -wedescribe the aitern-ate n-o-decoupier used in IWV’A. This i“sa pro- cAtief6r=b@ngihe~ 50Wg1~s.veiGeity-madGs $hat-o-ccurinnumw;eal fluid dynamim calculations that have velocities located at computational cell vertices. The basic idea of our pocedure is to detect and subtract, in each computtiionakdl, .velocity-.nmdes fbr. which the finite-difference approximations to the mean velocity and velocity gradients are. zero. ‘Tliiisis done in sucii a way that linear momentum is conserved. Since these modes have no caicuiateci “mean veiocity. or veiocity ~dEnt~.nQ.~.ytiml”form.s are~nt.rodueed ~ iiy.tkmme_calv; procedure. ‘I%e pZ0e4Zlrde*.in$20d~~ ~-SlllZiiintinleriC~l dtiH@ng- whose nature is discussed in Ref. 16. In typical hourglass velocity modes, the average velocity and velocity gradients at each cell center are zero when these are calculated from centered differences involving just the vertex velocities of the cell. Thus in the absence of an alternate node coupler, no force will be calculated to damp the hourglass mode. In order to detect the hourglass modes in KIVA, for each computational cell, we con- struct a set of vertex velocities with approximately the same computed mean velocity and velocity gradients as the velocity field in the KIVA calculation, but which does not have the hourglass modes. This velocity for vertex t’ of computational cell (i,j,k) is given by

M’ Un z mm U: = ‘z% ‘ m

where the summation is over the vertices m of cell (ijk); M’ ~ is the mass associated with vertex m;

are the velocity gradients calculated as in Appendix K; ~~~is the center-of-mass x-coordinate

M’ x x mm (L-2)

and ~~k and ~~kare center-of-mass y- and z-coordinates defined in an analogous fashion. It is seen that u~ is the sum of four terms. The first is independent of t’ and is the mass- averaged vertex velocity. The remaining three sets of velocities vary linearly in physical space, have mass-averaged velocity equal to zero, and, as we will show below, each has a derivative approximately equal to that of u~ in one Cartesian coordinate direction and derivatives equal to zero in the other two directions. It is natural to assume that velocity fields that vary linearly in space do not possess hourglass modes. The hourglass modes for computational cell (i,j,k) are then given by

anc _ n (L-3) ‘t ‘Ue– u’?.

138 ThequantityAJJC is auser-inputpa~etenbse-va~ue-is%~jca-]ly taken to be 0.05. A factor of+ is inserted into Eq. (L-4) so that the effect of our alternate node coupler con- forms.in some simple cases to that of previous node couplers.49 Itis easy ‘m ‘verify ~hdt-OUr~Od-e-COU@erconserves momentum. Indeedj hy. summing M’t8utanC over all vertices t and substituting from Eqs. (L-4), (L-3), and (L-1), one obtains

(L-5)

We now-show tha%.the.com~tidvdoe~ ty-derivatives ofbut~nc incell (i,j,,k-) areapproxi- mately zero. -The.finite=dffermw-ap~oximatims to -these vehe-ity der~v-ative-aregi~~en in.4ppendix-K-. ltcanb~enthat

where @l~zlg~ denotes.the.fi-nite. diffi%mceapproxirnation to the derivative in the x-direction in cell (ijk), ~1 and P2 are constant, and U1and uz are two sets of velocity values associated with the vertices of cell (i, ~,k).. Usin.~the.kfining.eq.nation~of.@UnC and Eq. (L-6), it is seen that in order to show

m-?)

[.:],,= [:],,. lJ lJ

Using Eqs. (L-1) and (K-2) -(K-4), we see that =— u l–lax ijk V;k [

aun +— (L-9) [ ?Y where

— u.. = cJk

the sums are over the faces a of cell (i,j,k), and Za,ja, and Zaare the arithmetic averages of the x-, y-, and z-coordinates of the four vertices bordering face a. By using the identity

~A~=O, (L-1O) a

Eq. (L-9) can be reduced to

Now ~a2aAa” i is a finite-difference approximation to the surface integral fs~k xi” dA, where S~k is the surface of cell (i, j,k). Applying Gauss’ Divergence Theorem to this surface integral gives

xi.dA= V o(xi)dV = V,jk . Is,, v.. LJk lJk

Hence we obtain

Similarly one can show

140 and

Using Eqs. (L-12), (L-13), and (L-14) in Eq. (L-n) gives

(L-15) which is the desired result. The analogous results for the derivatives in the y- and ~.diree-ti~~~–a~~btai ned ~~.M ~.. T.hu~the ~amputed .deri~~.ati%res~f5ut~~~in cdl(ijj~k) are nearly zero. Notallprevi= ous alternate node couplers have had this desirable property.. For example, consider. the- node.co.-.~ft~j.~~~~~~ ~.33Tc~w.pl~.t~r_pr~.~~a’ @ l-P=.a.t.l~d.i.~wpl~.iepra.~.ea~e-a~.a~.i~fi..~fi. which-the solution is invariant in-the k-direction, SALE-313 superimposes on the vertex velmities-u.kell (i;ej;k) a-veiocityfleldof the-forn_-

where.kmc = 14- + ua. — U~.- IUA..If the x- and “i-mmdinaie computation cells are rectangular, and if centered differencin~is used to calculate il(t3u~@/dx, the resuit”is

Thus [a(i5uu~c)/ax]~~is nonzero if this sum is nonzero.. Most..pre~iiMMalternate nwk couplers-have als.e had theundesirable property that they sometimes subtract hourglass modes even when these are not present. Consider a computed “veiocity fkid in which u = ax. By our assumption this velocity field has no hourglass nmdes... The alternate node coupler in CONCHAS-SPRAW8 subtracts a-velocity field where again Aa~ = U1 + U3 —U2 —u4. Hence Aaflc- xl + x3 — X2 — x4, and the algorithm superimposes a nonzero hourglass mode in cells where xl + x3 – X2 —x4 is not zero. A node coupler similar in spirit to ours has been proposed by Margolin.G4 His node coupler has in common with ours the property that the superimposed vertex velocities have no associated mean velocity or velocity gradients. His method, however, introduces undesirable hourglass modes when the cell vertices do not satisfy x 1 + x3 — X2 — x4 = O.

QUASI-SECOND-ORDER DIFFERENCING

In this appendix we describe the quasi-second-order upwind (QSOU) differencing scheme that can be used in Phase C to calculate convection. Many methodsG5~‘GS’7have been proposed for obtaining what are called monotone finite difference schemes. The QSOU method is a modification of a scheme proposed by van Leer,22 and the basic idea is perhaps best understood by considering the family of upwind differencing schemes for one-dimensional convection that are represented in Fig. M-1. In each scheme we assume the density profile within a cell is linear and has a value at the cell center equal to the computed old-time value of density for that cell. The old-time densities in cells i —1, i, i +1, and i + 2 are plotted using dots in Fig. M-1, and the density profiles in cells i and i + 1 are the solid lines. The four schemes of Fig. M-1 differ only in the slope used for the density profile within each cell. For each scheme, the mass convected across a boundary moving from point A to point B is the area under the density profile between points A and B. Because of this method for calculating convection, the new-time total mass in a cell is just the area under all the old-time density profiles between the new boundaries of the cell. Assuming the left boundary of cell i + 1 does not move, the new-time mass of cell z+ 1 is the area crosshatched in Fig. M-1. Before discussing the four schemes of Fig. M-1 we need to define some terms. A func-

tion p is monotone increasing (decreasing) if xl < X2implies P(XI ) < P(x2) [p(w)> p(zz)). A monotone function is one that is either monotone increasing or monotone decreasing. Now let xi denote the location of computed density pi. A difference scheme for convection is weakly monotone if it has the following property: if pi + In lies between pin and pi+2n and ~j+ 1‘+ 1 lies between Xjnand xi + Zn,then pi+ 1n+ 1 lies between pin and pi+2n. A difference scheme is strongly monotone if it has the following property: if and”

# ~ ~:ni-1 < ~.,n+l

then

(?> n+l> n+l> n h. p: ‘Pi+ ‘+1 ~ ~pf~~ S Pf+3 ‘pi –p~+l –?i+2 -P~+# . l%e-strong~y rnonutom-p~opert yisvery desirabie because strongly. monotonaschemesdb. ncd.have the undershoots and m7ershcmtsof-many higherordermethcxk.,2 1 Itis-easy to show that for the family of schemes of Fig. M-1, a suffkient condition for strongzmonotonicity. is the-fo-ilowing: if”pi ~ ~.a iies between pj~-”andj+. ~ ~R-j.tha.the.d~tity- pro~=s in .~dls i! i+- 13 .~.~~-.:.+ =->!-?.+-+---.“.. -.+.-flse+=k-uu~ “L.u. , .r~.rlyi. ~ .~.mAma.+=.-~.tiuLL1s=.=--uZ.l&i-G”n.‘-’e” Ca+”l“th”iscofi”dition”the-rncmOtimm- profiles ”condition: Lel-usnow.rcnsidar. each .of.the schemes of Fig. M-1. In donor cell -differ~eingj the ~~ape%%-;tki ~ =~-~ c‘+1 is-take~’~-~be=zem~ ~ano~mildifiemneing satisfies tltemmmtone= profiies-condition and hence is strongi~. monotine.. Dbrmr.celldifferencing G first-order accurate in space, however, and has too much numerical diffusion for many applications. To remedy this accuracy problem, one might consider the centered-gradient scheme ofl?ig, M-1b. H=eth~-lop*pl~x-=within eaeh ~Al is given-by

!zi+l–!?i=~ y~ (M:~j axli= %.. ‘

T.hi~~.keme is ~tab~e-a~d=se~~~d.~~~~accum~e; but-not- weakly monotone. “Tlfiereason is shown in Fig. M-lb. The density profiles in regions of large change in i@/tkrcan have large undershoots m overshoots.. In Fig. M-1b the density profile-in cell i+- 1 has-values less tlmm:p$ and-henc~p~+ ~‘+ ~,which -is=theaverage vaiue of’p und~~ihec~hatched-’a~ea- of the profile, is less than pin. Va- =Leer’s-scheme remeets this problem-by limiting-—t]~e-~r~~itu~-of-tbe siope-.zz-If p~=-ii~s-between pi _ 1‘and”pi + ~‘i, then the magnitude of the sibpe ~p/azli iS required-to be small enough that the density profile in cell i assumes values between pi_ ~n and pi+ ~_‘. If pin does not lie between pi_ ~~andpi+ ~~,then-the slope ~p/~x!jis taken to be zero. The resulting change in the slope in cell i + 1 is shown in Fig. M-lc. The slope in cell i is not lirnitedl%ca-use-the density profile in this cell already assumes values-between pi_ ~_nand ~;+~=..ln ‘tcan be-&omn=that=xJ.a~~le&~.heme-isweakly-monotone and is-sewmd-mxi~ accurate for computed densities for which the slope-limiting. procedure is not used. The

1-43- ●

a. Donor cell b. Centered-gradient

w ●

i—1 i i+2 I 1 1 I

B A B A

c. van Leer d. QSOU

●

i—1 i 1 i+2 I t {

CDBA B A

Fig. M-1. Density profiles for a family of upwind convection schemes. 144 SC&YIldS=11.QkSl’KCM@;llMM3.0tQEMkFOE=Siill~l~j iftbf) l~ftb~tllld~~’ Qf”L~ll’i”iS~GWd’flW= pointC’~ point-12 -inFigi M-le, then we=would-have p;!+ 1->-p;..-,.-.~, n+ 1-evemthoqlr the-” values-~i _ 1‘; p~~~~pi+1‘j amd-’p~+ A%are monotone increasing. Tine QSOU scheme was devised ti.satisf”y themonoti~WQfilK mnditidn.and.hen~ ‘w be Strongly mmmtuna T-h@SObT sckeme-mig~t-also be called the minimum gradient scheme because if pin lies between pi_ ~~and pi+ in then the slope is taken to be

(M-2)

Aswiti.vanhr%dwne,.if ~.ifldORSmd ~i~h~t~-e~~.-~i_ .I-n.a-d ~i+ ~rzv.....+h~n..~A /A....lt is $a.kep. to bezero. In the~xamp2e ofFig. M-ldj the slopes in-cells iand-i-+- 1-am-’mth-limited’oy- thi s-TIwse.~1p%icm. We now show the-QS-CllIprescripticm f~r.d@d~{isatisfies the monotone-profiles ccmdition. Suppose pi_ IV .S Pin S Pj.+~.‘“Jthe monotone-decreasing.case is handled similarly. ‘Tiien: [email protected]~,.

and

(M-3)

From (M-3) we see that the density profiles are monotone increasing within cells i – 1, i, and i +1: It ~emains-tx-slmw that

ap Ax (3PI Ax Pi< + u til~.l;

lApil lAP,_l] sign (Api) rnin — — if Api Api_l >0 ap Ax. ‘ Axi_l ) — ( L (M-5) Zi, – d 0 if Api ApL_l <0 . Then the density pa, used for fluxing through cell face a between cells i and i +1, is given by

(M-6)

In Eq. (M-6) Xais the location of cell face a, 6Va is the flux volume, taken positive if cell i is the upwind cell, and Vi is the volume of cell i. There are many possible ways to extend this method to a three-dimensional mesh of arbitrary hexahedrons, and we have chosen a simple extension in which the fluxes in each coordinate direction depend only on gradients in that coordinate direction. Consider the determination of the quantities pa”, where p is one of the cell-centered quantities to be convected (p = pm, pI, pk, or pk3/2/8), ~is the convective subcycle number, and a is the common face between cells (i, j,k) and (i+ l,j, k). Fluxes in the j- and k-coordinate directions are treated analogously. Using a straightforward extension of (M-5), we first calcu-

146 late the cell-centered derivatives-of p.withre~ect.ti.ddmce.s.in.the~.c~mdinate direction:

Ir@vf-7 );

where the Cell-centered Iwatimwx-c jkcare~-aleulated -usingEq. (55) and the new-time vertex locations. Then pa~is obtained from an extension of (M-6):

.“. ~-+r [ ill X;j,kl 1 – — if tiVa >0 a— , L,JJ?1X ( Vv ) LjA (M-9) 1. _.@v

‘f+ l,j,k [ z Ii+l,j,k lxa-x:+ljkl[’ + v~j,k~ ‘f 6va

IAu:I \Au:-ll ,~ AUV Auv sign min — — “ ~_l>o au v (Au;) 1 — (lAxil ‘ lAxi_ll ) (M-9) ~ijk– 0

In (M-9)

Au: = U* —Uv. i i- I,j,k l,J,k and

n+l Axi = X:;; j,k — x., l,J,k “

When vertex (i,j,k) lies on a computational boundary, then the derivative, with respect to distance in the coordinate direction going into the boundary, is taken to be zero. In order to calculate u~~by extending (M-6), we also need a quantity analogous to 6Va/Vi. In a one-dimensional calculation with constant cross-sectional area, this quantity is the Courant number based on the fluid velocity relative to the grid velocity. It is convenient to base its counterpart for momentum fluxing on the ratio of the mass flux (6Mpc)V, defined in Sec. JILG, to an open-flow-area mass (M@V that is defined by

(M-1O)

148 The mass flux (bikf~c)’ is taken to be positive if mass is being added to vertex (i+ l,j,k) by the-mo~ematfi~~~ jn.which~a~ .nmmentum .c~ll (z,j,k) is the upwind cell. If.@M#).? i%itegZdiV~Yi~-~-e~~~-c”e-il’.(’i+ i’YJ“;@ ‘is t’ih~‘Up~ind’ceii.

APPENDIX N

ITARZITAL .IJCHWJR.CELL .DIFFE REIYCING-

Here we describe the partial donor cell differeneing procedure that can be used to evaluate the cell:f”ce quantities~~~, where Qs%a-nds-fimany o~thevaria-bl~~ pfi, pI~ph-, pL, or u. We first describe the procedure for the cell-centered quantities pm, pf, pk, and pL. Letthe regularee]l in-question-be c=lled-ce-ill; arlWlet~he-nei@b~ring;cell that is common tn the face a be called cell 2.. The quantity QQ! is-valuated as-an upstream- weighted average. ofQ~~ and Q~~. The firststep is-therefor~to determine which ~mllis the ‘Upstreamordimm%eell; ~-. m=- =i=~lt%d~---- Merrnified””’by the -sig~ i3f”3-V~if”tixi~>- O;.thell-ceil 2 is the upstream or donor cell ”wliile ceil “l-ii the-dbwnstrearn-or acceptor cell, and vice versa. Tiius we define

v. ~: =\’Q2. ‘f ~v= > g 1 Q; if i5Va < 0 (N-1) and

,Q~ if 8V” > 0

Q:={ ‘- -

Q; if 6VG <- 0, 149 where the subscripts “d” and “a” refer to “donor” and “acceptor” respectively. The partial donor cell prescription for Q.V is then given by

(N-2) Q~=4Qj(l+aO+900+*Q~(l–aO– POQ, where ao and POare adjustable coefficients (O—< ao + ~oC —< 1), and

216VJ c= (N-3) VI + V2 is an effective Courant number based on fluid speed relative to the mesh. If ao = PO= O,the above prescription reduces to centered differencing of the convective terms, which is unconditionally unstable unless compensated by a sufficient amount of diffusion. If ao = 1 and PO= O,pure donor cell or upwind differencing results. This scheme is stable, but is too diffusive for most applications. Its effective numerical diffusivity is # UIAXin the x-direction, &lulAy in the y-direction, and # WIAZin the z-direction. If ao = Oand PO= 1, the so-called interpolated donor cell scheme results. This is effec- tively a weighted average of centered and donor cell differencing, with the weighting factor set at the value for marginal stability. This scheme is less diffusive than pure donor cell differencing but is not monotone (see Sec. DIG). The optimum values of ao and Doin any particular calculation must be determined empirically, usually with reference to the nominal values ao = 0.1, Do = 1 as a starting point. To prevent the possible development of negative turbulent quantities, the values ao = 1, PO= Oare used instead of the input values when Q = pk or pL, That is, the con vective transport of turbulent energy and length scale is always done by pure donor eel” differencing. The procedure for evaluating upon the momentum cell composite faces is entirely similar but is based on Nllpc instead of 8Va. Let the vertex in question be called vertex 1, and let the neighboring vertex which shares face j3be vertex 2. Define

u; if (M4jv > 0 v— ‘d – [ u; if (M4~)v < 0 (N-4) and

u; if (t3it4jv > 0 u: = { u; if (&14jv < 0 . 150 Then uawis given by

VFFrf.m’

‘whmva-, b; e; and-d-refer’mthe fcmr-reg@ar cells-sharing the regular cell edge joining vertices 1 and 2.

T-hePhase C mm~rir~g-al~wtithm-mimics the acivection terms of”the Navier-SlOkefi equations and consequently suffers from the usual numerical difficulties associated with. those terms. T-he womktwo problms-are-numerieal diffuskm and -dispersiontmrmation- errors. The first of these is well known, and the KIVA program allows the user the option of using interpolated donor cell differencing or quasi-second-order upwind (QSOU) differ- ~nc.ing.to .redu~~this-a-rtifl~-ialsmoothing’~ more or less ae~~ptable levels ‘ihile--retainin~- stability. Dispersion errors can be significant when using interpolated donor cell-diffem- ~_he.~ia@G ~ge..c.;harae.teri~$;~--i~.spa%i:a;-: elmi.n~ [email protected]*iutien-J“+ oseilla%ions-’,vith -a-period-of’a%veml cells; ‘wbieh %em=to%e-mostsevere-irr re@cmsoi%tee~- gradients such as shock waves or flames. These oscillations can be severe enough to drive species masses negative in some cells, and it is this behavior that the Robin Hood (RH) algorithm was designed -’~ prevent: If tl~e-m-levant-~antities-are posi tive in a cell; it is unaffected by RHj and “ItH-isnot needed, and therefore not used; if QSOU differencing is used. RH is a primitive form of the flux limiter that derives its name from the fact that it steals from rich cell~-md@tiesto. the.wm.. STupposethat.a .~lven .~dl has a neg+tive species mass. RI-Isearches.the.six facing. neighb.oring.cells and picks the one with.the lZmgpst-mass-of’tiie particuiiar species. Einou.qtispecies mass is removedlrom.thti~ejlta- ‘tifigits .paarmeighbaraptO--T~To-;ma3sforthatspec-ies= Tine s~eeifle-. internal ‘energies-and’ turbulent kinetic energies and length scales of the two cells must be adjusted for this mass exchange. If the rich cell does not have enough species mass to bring its neighbor up to zero, all of the mass of that species is transferred to its neighbor. In this way, negative values are reduced without introducing new ones. While the RH procedure may seem ad hoc, it probably does more good than harm. The diffusion that it represents is typically localized in space and time, and it does reduce the magnitude of dispersive errors slightly. Most importantly, it represents a simple attempt to limit convective fluxes to physically realistic values, maintaining positivity in quantities that must be positive to allow calculation of other quantities such as chemical reaction rates.

APPENDIX P

ANGULAR MOMENTUM CONSERVATION LOGIC

In an axisymmetric swirling flow with free-slip boundaries, the total angular momentum should be conserved. However, in their basic form the KIVA difference approximations to the momentum equations simply conserve the three Cartesian components of momentum, and this does not imply angular momentum conservation because of truncation errors. This lack of conservation is a serious problem for calculations of swirling flow in internal combustion engine cylinders because the truncation error effects are typically larger than the legitimate physical swirl decay due to boundary layer drag. In practice, only the truncation errors in the rezone calculation are found to be significant. We have therefore devised an optional angular momentum correction procedure which is incorporated into the rezone calculation of Phase C. The essence of the procedure is most easily explained in the context of a model problem, namely the pure Eulerian convection of momentum represented by the differential equation

a (PU) —+v. (puu)=o. (P-1) al

Consider the augmented differential system

a(pu) — + V.(puu) = ~[(y–Yo)i – (X– zo)jl , (P-2) at 152 *psj — + T.(p_suj= O , dt

s=(x_xGju–(y_–yJu, (P-4)

. . ‘w&e.kisAu~-’E-regaded--aa-5y.s&.m:Qf:?f’?.ve “eq=z.”n%xw.-fk+k fkdefwdez% .Vi+rla’b:ks.Zi, v> w, ~, ands. Equation (P-4), which shows thats is the angular momentum per unit mass aboutth~ax-is4x@ = {x~,y~); plays-th~role efa-~wnstraint’which implicitly determines . ..—.— the depenc.kmt”variabie ?”= ~(x,t). A-n equation for ~ “canbe obtained “by combmmg~ (P-2)-(P-4j; andwhen tihisis done one finds that ~is identically zero. Thus the system of- Eqs. (P-2) and (P-4) is precisely equivalent to the origjnal Eq, (P-l). ‘The idea now- is to write di ff~mtme---a~xitiati~nsti=ti~qs;s; @2)-(P=4) instead “of”the equivalent Eq. (P-1). These difference equations will of course not be equivalent to those datained.by.naively -differencing .Eqg(P-1) because oftrumeation errors, 13ifferenein~ (lZ~-(P-4) ii pref~rable because this system .explic-itl~r-containsac~nser~~atlon equation for-t“heangular momentum d“ensitys; am% aco~enati-ve-dlT6renci-n-g-of=thi”s equation will automatically conserve angular momentum. .A.suitable difference scheme fcwEqs~ (P-1) -(P-4)is-

(pu)n+l – (pU)” ~+lrLv__ ~L_)~=—(Z.- .-=~.s. z + (PL!&# = ~. -v w) iL At “o

~n+l (psJ — (psln + -(pstl)n =0, At

n+l , n+l , .n+l .$. =- (z:— -Z.O)B.. _.@_ydQ. , where -k the s@ki] diffe~nceoperdto~se~~o-approximate the dirfferentiai “d.iver~ gemce operator. T%is scheme may”be rewritten as

(p:gj (pH)’r+i =-pm+i~-+~t.pn+ i[(y.–.yo~=*– (~-– %;)jl ,

(ps)‘+l=(pS)n –A~KV. >(pSU), (P-9)

n+l=(x– xo)v”+l–@y)u”+l (P-lo) s 0’ where ii is the value.of u~ + 1that.woultieAstf lmEq. .(P.-5).if.~~+ 1 were ~ro= This of. ,m~me-is-the-:.zalueOfti~+~=tbat-.voul#we-Obtain&%y~imply diIffere~mingEq. (P-1>. Z% vertic=l Ccmqmnent-oi%q. (I%) isjust-

153- (P-11) W=n-i-1 ii, which shows that the procedure does not affect w, as one would expect. The component of Eq. (P-8) in the direction of (x – xo)i + (y – yo)j is

(P-12) (x–xo)un+l +(’y-yo)un +l=(x-xo)u +( Y-yJ;.

We now observe that Eqs. (P-1O) and (P-12) are two equations for the two unknown quantities U.fl+1 and u~+ 1 in terms of s~+ 1, which is obtained from Eq. (P-9). Notice that it is not necessary to explicitly solve for ~n+ 1. The solution of these equations is easily found to be

u ‘+1 = d-l[(x_xo)%+ (x _xo)@–yo)fi -@-yo)s’’+ll , (P-13)

u“+l=d-l[(x –zJ@–yo)fi +@-yo)2; +(x-xo)s”+ll, (P-14) where d = (x – XO)2 + (y – YO)2.Once u“+ 1and un+ 1 have been determined, s~+ 1 is.of no further interest and need not be retained. The net result of the procedure maybe described in the following way. One first calculates a provisional value of u~ + 1 using the basic difference scheme that one would adopt on other grounds, without regard to angular momentum conservation, and one further calculates sn+ 1 using the same scheme. One retains the vertical and radial components of this un+ 1but discards the tangential or azimuthal component and replaces it by the value that agrees with sn+ 1. Described in this way, the procedure sounds ad hoc and unjustified, but the preceding development shows that it is in fact a well-defined and consistent difference approximation to the differential problem, Application of this procedure to the rezone phase of KIVA is simple and straightforward. At the start of the rezone phase, the vertex-centered quantity s~~ is initialized in terms of the Phase B velocities using Eq. (P-4) above:

(P-15)

The angular momentum density s~~’+ 1 is computed by replacing u~k with sijk in Eq. (127) of the main text, so that angular momentum is fluxed in the same way as linear momentum. The corresponding values of u~k~+ 1and v~k~+1then serve as the provisional values ili.jkand ~~k, and the final values of u~k~+ 1and u~k~+ 1 are obtained from Eqs. (P-13) and (P-14) with x and y replaced by x~k~+ 1 and y~k~+ 1, u and u replaced by ii~k and fi~~,and

154 W+ 1 and u~+ 1 replaced by u~~v+ 1 and u~~v+1. The intermediate vertex positions are @Yen=kly=

X;k = [(NS– V)x!?k+ vx;: lYNS , wh-em-NS is-the number of convective subcycles.. ~e.corr~.ti~n.pQmdure.i~.optional and is activated -byan input-flag. It is-al~vay-sd-e-ti~’atd ifinputflagCY-L is zero.

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